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Enormous amounts of mass required

I'm not going to really address your first question of whetherhow an ubermassive black hole could exist. Within our current understanding of gravitational physics, black holes are formed by stuffing enough mass-energy together in one spot, and it doesn't sound like the humans of your story are rounding up all the stars. I'm going to assume for the sake of your story that a black hole is formed in some exotic manner that massively violates conservation of mass-energy, such as by somehow harvesting the zero point energy of the universe, or manipulating the cosmological constant (a kind of energy density of empty space), or something even whackier.

Black holes are not vacuum cleaners

The 1000-Sun mass isn't arbitrary, since it's just about the minimum mass to really start affecting the closest 1000 stars, and even then quite languidly. By a rough estimation, a black hole of 1000 Solar masses is sufficient to gravitationally dominate a roughly 45-ly radius, such that objects within that 45-ly radius would orbit the black hole, not the galaxy center (like the Moon orbits the Earth, rather than following its own independent orbit around the Sun). This is from a quick calculation of the Hill sphere, and the gravitionalgravitational mass of the Milky Way. A radius of 45 light-years roughly encompasses the nearest 1000 stars, which should be a proxy for the number of planets, within a factor of 10 or so.

There's enough of that digression. If you want destruction, how about wewe'll exit the Newtonian regime?. Let's make the event horizon reallyreally big, so all the stars are close to it. The Schwarzschild radius is, $r_S=\frac{2GM}{c^2}$. To have an event horizon 4.22ly in radius, reaching the nearest habitable planet, we need a black hole of 4 times the mass of the Milky Way as a whole, at $2.688*10^{43}$ kg. This will instantly swallow Proxima Centauri, and the rest of the 1000 closest stars should fall in within about 230 years.

 

Alternatives to black holes

 

I'm not going to really address your first question of whether an ubermassive black hole could exist. Within our current understanding of gravitational physics, black holes are formed by stuffing enough mass-energy together in one spot, and it doesn't sound like the humans of your story are rounding up all the stars. I'm going to assume for the sake of your story that a black hole is formed in some exotic manner that violates conservation of mass-energy, such as by somehow harvesting the zero point energy of the universe, or manipulating the cosmological constant (a kind of energy density of empty space), or something even whackier.

The 1000-Sun mass isn't arbitrary. By a rough estimation, a black hole of 1000 Solar masses is sufficient to gravitationally dominate a roughly 45-ly radius, such that objects within that 45-ly radius would orbit the black hole, not the galaxy center (like the Moon orbits the Earth, rather than following its own independent orbit around the Sun). This is from a quick calculation of the Hill sphere, and the gravitional mass of the Milky Way. A radius of 45 light-years roughly encompasses the nearest 1000 stars, which should be a proxy for the number of planets, within a factor of 10 or so.

There's enough of that digression. If you want destruction, how about we exit the Newtonian regime? Let's make the event horizon really big, so all the stars are close to it. The Schwarzschild radius is, $r_S=\frac{2GM}{c^2}$. To have an event horizon 4.22ly in radius, reaching the nearest habitable planet, we need a black hole of 4 times the mass of the Milky Way as a whole, at $2.688*10^{43}$ kg. This will instantly swallow Proxima Centauri, and the rest of the 1000 closest stars should fall in within about 230 years.

 

Enormous amounts of mass required

I'm not going to really address your first question of how an ubermassive black hole could exist. Within our current understanding of gravitational physics, black holes are formed by stuffing enough mass-energy together in one spot, and it doesn't sound like the humans of your story are rounding up all the stars. I'm going to assume for the sake of your story that a black hole is formed in some exotic manner that massively violates conservation of mass-energy, such as by somehow harvesting the zero point energy of the universe, or manipulating the cosmological constant (a kind of energy density of empty space), or something even whackier.

Black holes are not vacuum cleaners

The 1000-Sun mass isn't arbitrary, since it's just about the minimum mass to really start affecting the closest 1000 stars, and even then quite languidly. By a rough estimation, a black hole of 1000 Solar masses is sufficient to gravitationally dominate a roughly 45-ly radius, such that objects within that 45-ly radius would orbit the black hole, not the galaxy center (like the Moon orbits the Earth, rather than following its own independent orbit around the Sun). This is from a quick calculation of the Hill sphere, and the gravitational mass of the Milky Way. A radius of 45 light-years roughly encompasses the nearest 1000 stars, which should be a proxy for the number of planets, within a factor of 10 or so.

There's enough of that digression. If you want destruction, we'll exit the Newtonian regime. Let's make the event horizon really big, so all the stars are close to it. The Schwarzschild radius is $r_S=\frac{2GM}{c^2}$. To have an event horizon 4.22ly in radius, reaching the nearest habitable planet, we need a black hole of 4 times the mass of the Milky Way as a whole, at $2.688*10^{43}$ kg. This will instantly swallow Proxima Centauri, and the rest of the 1000 closest stars should fall in within about 230 years.

Alternatives to black holes

 
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NonstandardDeviation
NonstandardDeviation
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I'm not going to really address your first question of whether an ubermassive black hole could exist. Within our current understanding of gravitational physics, black holes are formed by stuffing enough mass-energy together in one spot, and it doesn't sound like the humans of your story are rounding up all the stars. I'm going to assume for the sake of your story that a black hole is formed in some exotic manner that violates conservation of mass-energy, such as by somehow harvesting the zero point energy of the universe, or manipulating the cosmological constant (a kind of energy density of empty space), or something even whackier.

The other problem is that black holes are surprisingly nondestructive. A black hole, once formed, doesn't really suck things in to their doom. Let's consider a relatively tame 1000-Sun black hole.

In the Newtonian regime (far away from the event horizon) black holes behave like any other mass, and objects will orbit around them as with any other mass. The Sun is heavy, but for example, rocks from outside the solar system tend to fall inwards, whip around the Sun, then head back out to deep space. Objects have a certain amount of potential energy relative to the central mass they're orbiting, and barring extraordinary events such as collisions, this energy is conserved. Unless an object is aimed directly at the central mass, i.e. it has nearly zero angular momentum, it won't get sucked in. If you were to magically plop this black hole in interstellar space, yes, things would be pulled, but by conservation of angular momentum and energy these things (stars, planets, comets, etc.) would end up either slingshot away from the black hole, in orbit of it, or relatively unperturbed further away. A black hole of 1000 solar masses would make a mess of stars' orbits around the galaxy, but each solar system would stay intact, and life would go on. If you're wondering about how many stars would be sucked in, I would estimate none. Space is very large and even 1000-Sun black holes are small, and it is really easy for an infalling star to miss the black hole and whip around back into deep space far away. Also, it would take objects about 10 million years to fall in from 20 light-years away, so this wouldn't exactly be a pressing issue.

The 1000-Sun mass isn't arbitrary. By a rough estimation, a black hole of 1000 Solar masses is sufficient to gravitationally dominate a roughly 45-ly radius, such that objects within that 45-ly radius would orbit the black hole, not the galaxy center (like the Moon orbits the Earth, rather than following its own independent orbit around the Sun). This is from a quick calculation of the Hill sphere, and the gravitional mass of the Milky Way. A radius of 45 light-years roughly encompasses the nearest 1000 stars, which should be a proxy for the number of planets, within a factor of 10 or so.

There's enough of that digression. If you want destruction, how about we exit the Newtonian regime? Let's make the event horizon really big, so all the stars are close to it. The Schwarzschild radius is, $r_S=\frac{2GM}{c^2}$. To have an event horizon 4.22ly in radius, reaching the nearest habitable planet, we need a black hole of 4 times the mass of the Milky Way as a whole, at $2.688*10^{43}$ kg. This will instantly swallow Proxima Centauri, and the rest of the 1000 closest stars should fall in within about 230 years.

If I were you, I'd take an alternative path, since the physics is already being bent in knots. How about Modified Newtonian Dynamics?. MOND theories and other alternatives to general relativity were somewhat recently proposed as alternatives to dark matter and dark energy. You don't have to understand this math particularly well, especially since it's a quite abstruse field of active research.

My suggestion for the sake of your story is that you explain your humans' FTL-cheating tech as exploiting a localized deviation from general relativity that encompasses our 500-2000 star systems. Unfortunately, the FTL cheat only makes these extra terms in the equation bigger and bigger, so that after many uses this pocket starts collapsing in on itself. Where today the cosmological constant is positive, driving expansion, it gets altered locally to more and more negative values, causing contraction.

Another possibility is that where Newtonian gravity is purely $1/r^2=r^{-2}$, FTL drive use starts to add a $1/r=r^{-1}$ or constant $r^0$ term - if attractive, then planets and moons start drawing in closer; if repulsive, everything starts drifting away.

If you have any more questions, I've done more math I didn't bother writing up and would be glad to share.

If you'd like to play around with gravity, try this toy. As a warning, it's not to scale, but you can try spawning a ring of dust in a circular orbit around one star, and then adding a second star to see how most of it is kicked out of the system.