SHORT ANSWER You probably need to make the system contain a white dwarf star orbited by a gas giant planet and by a Earth-like planet orbiting just slightly farther than the gas giant planet. Due to the extreme improbability of surviving planets in the habitable zone of a white dwarf, that star system would probably have to have been created by highly advanced aliens for some reason.
LONG ANSWER explaining the reasoning.
I would suggest that you make the small bright star as small as you plausibly can and replace the larger dimmer star with a gas giant planet or a brown dwarf with a much greater radius.
Thus, if the giant planet or brown dwarf is wider than the star, its shadow will expand with distance becoming wider and wider with increased distance from the star. And if you make the giant planet or brown dwarf very close to the star, compared to the diameters of the star and the dark object, the shadow of the dark object may cover a significant percentage of the orbit of any planets orbiting farther out - a percentage equal to that of the diameter of the planet or brown dwarf compared to that of its own orbit.
If the planet to be eclipsed orbits in exactly the same orbital plane as the inner planet, the length of its eclipses should be the same percentage of its orbit as the diameter of the inner planet or brown dwarf is of its own orbit.
If the gas giant planet or brown dwarf has a diameter of 0.5 percent of it's orbital circumference, its shadow should be 0.5 percent of the orbital circumference of the outer planet, which thus would be in eclipse for 0.5 percent of its year.
So if your eclipse by the inner planet or brown dwarf lasts for five Earth days, for example, and is 0.5 percent of the outer planet's year, for example, the total length of the outer planet's year would be 1,000 Earth days or about 2.737 Earth years.
But that is assuming that the inner planet or brown dwarf is motionless and only the outer planet is moving.
If, on the other hand, the inner planet or brown dwarf orbits the star and the outer planet is motionless, the eclipse would have to last for 0.5 percent of the inner planet's year. So for the eclipse to last for five Earth days, for example, the year of the inner planet or brown dwarf would have to be 1,000 Earth days or about 2.737 Earth years long.
But both these calculations are wrong, because both the inner planet or brown dwarf and the outer planet have to be moving in orbit around the star to keep from falling into it.
Because the inner planet has to be moving faster than the outer planet, the inner planet will catch up to the outer planet and eclipse it for a while before pulling ahead of it.
In order for an eclipse of one planet by an inner planet to last as long as possible, the two orbits must be as close as possible to make the relative orbital speeds of the two planets as close as possible.
Because the planets in our solar system have absolutely and relatively widely spaced orbits, each planet in our solar system has a significantly wider orbit and slower speed than the next innermost planet. Thus each planet is overtaken in orbit and passed by the next innermost planet in a short time.
Astronomers always assumed that would be the rule in other star systems, until exoplanets were discovered and it was found that many solar systems are very different from ours, including some star systems where the planets orbit very close together both relatively and absolutely.
The smallest absolute difference between two exoplanet orbits is between Kepler-70b and Kepler-70c. Kepler-70c orbits only 0.0016 Astronomical Units, or about 240,000 kilometers, or about 149,129 miles, farther out from Kepler-70 than Kepler-70b.
The smallest relative difference between two exoplanet orbits is between Kepler-36b and Kepler-36c. Kepler-36c orbits only 11 percent, or 0.013 Astronomical Units, or about 1,944,772.3 kilometers, or about 1,208,425 miles, farther out from Kepler-36 than Kepler-36b.
https://en.wikipedia.org/wiki/List_of_exoplanet_extremes1
So if planet c orbits 11 percent farther from their star than planet b, their relative orbital speed should be easy to calculate. I believe the relative orbital speeds of planets b and c would be proportional to the square route of a constant divided by the radii of their orbits, and so if the orbital radius of planet b is 1.00 and that of planet c is 1.11, their relative orbital speeds should be in the ratio of 1.00 and 0.949 if my calculations are correct.
In order for the bright star in the system to be very small in diameter, significantly smaller than a gas giant planet or a brown dwarf, it will have to be a very dim red dwarf.
The dimmest red dwarf star that I know that has planets orbiting in its habitable zone is TRAPPIST-1, a spectral type M8V or M8.2V. TRAPPIST-1 has a radius of 0.121 plus or minus 0.003 that of the Sun. The Sun has an equatorial radius of about 695,000 kilometers and thus a diameter of about 1,390,000 kilometers. So TRAPPIST-1 should have a radius of about 84,095 kilometers and thus a diameter of 168,190 kilometers.
https://en.wikipedia.org/wiki/TRAPPIST-12
The largest planet in our solar system, Jupiter, has an equatorial radius of 71,492 kilometers and thus an equatorial diameter of 149,984 kilometers. Thus if Jupiter orbited TRAPPIST-1 its shadow would be a cone that tapered to a point, but it would do so very slowly.
If a gas giant planet was a little bit more massive than Jupiter it would be a little bit bigger than Jupiter. But making a gas giant planet a lot bigger than Jupiter would make it smaller and denser than Jupiter. Jupiter is near the largest size possible for a gas giant planet.
That is, Jupiter is close to the largest size possible for a cold gas giant planet orbiting far from its star. astronomers have discovered many "hot Jupiters", gas giant planets orbiting close enough to their stars to receive as much radiation as Earth does, or even much closer to their stars, close enough to have temperatures much hotter than Earth's. And the high temperatures of some "hot Jupiter's makes their gases expand and swells up the diameters of those planets.
The list of largest exoplanets includes several which are two, or three, or four times the radius of Jupiter. https://en.wikipedia.org/wiki/List_of_largest_exoplanets3
But if your planet that is eclipsed is orbiting in the habitable zone of its star, and if the gas giant planet that eclipses it has to have an orbit almost identical in size in order to make the eclipses last as long as possible, the gas giant planet should be only slightly hotter than the habitable outer planet and thus it might not be swollen up much due to heat.
Brown dwarfs are objects more massive than gas giants and less massive than stars, with masses ranging from about 13 times that of Jupiter to about 75 to 80 times that of Jupiter. Because of their greater mass, brown dwarfs are many times as dense as Jupiter and so have similar diameters, rarely getting much larger.
https://en.wikipedia.org/wiki/Brown_dwarf#Superlative_brown_dwarfs4
Thus the radius of the red dwarf star should be decreased to make it even smaller and dimmer than TRAPPIST-1.
Star EBLM J0555-57Ab has a radius of about 0.84 that of Jupiter, and thus about 60,0853 kilometers, and a diameter of about 120,106.5 kilometers, smaller than a gas giant planet the same size as Jupiter.
https://en.wikipedia.org/wiki/EBLM_J0555-575
Possibly the red dwarf star could be replaced with a white dwarf star.
The smallest white dwarf star is listed as GRW +70 8247, with a radius 0.005 that of the Sun.
https://en.wikipedia.org/wiki/List_of_white_dwarfs6
GRW +70 8247 has a radius of about 3,478.5 kilometers, or a diameter of 6,957 kilometers, making it smaller than Jupiter, and even smaller than all the planets in our solar system except for Mars and Mercury. So a Jupiter-sized,or even Earth-sized, planet orbiting a white dwarf star the size of GRW +70 8247 would cast a shadow that got wider and wider with increasing distance from the planet.
Of course highly advanced and powerful aliens wopuld probably have to assemble that star system, bringing in the inner and outer planets from other star systems to orbit that star, since it would be unusual for planets, especially a habitable planet, to naturally orbit around a white dwarf star.
PS see my answer here: What are the day and night fluctuations for a moon orbiting a planet the size of Jupiter?7
