Now, forgive me if I am missing some things, but what you are looking for in this post does not seem to be that challenging. I'll propose a solution with near-future technology.
Summary
- Use two large-ish asteroids in the belt in locations that make them unlikely to be hit by space debris.
- You can't use a space station due to vibration problems, you need to build into an asteroid with some serious mass.
- Do not use the interferometer for objects in the plane of the solar system, instead use it for objects closer to the axis of rotation of the solar system.
- Use multiple telescopes/apertures to get the spectrum you desire
- Coordinate image taking with a constellation of navigation satellites
- Aggregate and post-process information later
Sources
On the asteroid belt's orbital and size distribution. Gladman, B., et al., 2009.
The application of interferometry to optical astronomy imaging. Baldwin, J. and Haniff, C., 2002
Integrated optics for astronomical intereferometry, Part I. Malbert, F. et al., 1999
Integrated optics for astronomical intereferometry, Part II. Berger, J. et al., 1999
Integrated optics for astronomical intereferometry, Part IV. Berger, J. et al., 2001
Integrated optics for astronomical intereferometry, Part VI. LeBouquin, J. et al., 2005
Astronomical Interferometry on the Moon. Burke, B., 1985
Method
Site selection
The asteroid belt is relatively sparse. Estimates of the number of asteroids over 1 km range from 1 to 2 million. Gladman, 2009, finds the power law scaling of asteroids with size at in this range to be -2.5; so the number of asteroids is $N \propto r^{-2.5}$; this would put our 100 m asteroid estimate at 300-600 million.
The inner part of the asteroid belt is distributed between approximately 2.2 and 3.3 AU from the sun, at an inclination up to 20 degrees. This corresponds with a torus with major radius 2.75 AU and minor radius 0.55 AU. This gives a volume of about 16 cubic AU, or $5.5\times10^{25}$ km$^3$.
For an assumed 500 million asteroids of 100m or more, this gives a density of $9.1\times10^-18$ km$^{-3}$; or, assuming a random distribution, an average distance between objects of 500,000 km; more than the distance from the Earth to the moon.
For objects over 100 m diameter, the density is as low as the density of moon-sized objects near the Earth. Since the Earth is in no great danger of being hit by the moon, our interferometer is not in particular danger of being hit or otherwise affected by another asteroid. For objects smaller than 100 m diameter, these are approaching the size of objects that we do move in space. If we are able to bring a large telescope installation to the asteroid belt, we should be able to deflect an asteroid of this size.
Vibration management
A space station will not have the optical resolution required due to vibration. I share some vibration information from ISS in this post. The sort of vibrational stability needed to resolve a milliarcsecond with a 100m baseline receiver is aobut 0.5 $\mu$m; the ISS vibrates with an amplitude of about 4 mm.
How can we get a stable enough platform? Well, the Earth is obviously stable enough for giant interferometers like LIGO. The asteroids we need to pick will be intermediate in stability, since they are between the size of the Earth and the ISS. I could not find reasonable information or calculations to perform regarding the stability of a platform built on or into an asteroid, but we will assume that an asteroid must be selected with the proper characteristics of a stable platform. I would imagine we would chose an asteroid of 1km or greater diameter, if possible. The larger, the more stable.
Directing the interferometer
If you have two points on opposite sides of the asteroid belt, then it makes sense that you will not be able to resolve the objects which lie in the plane of the solar system. There will be too much interference from other asteroids, or the sun or what have you.
The solution is simply to restrict your observations to one or the other hemisphere. For example, you can build your telescopes on one side of the asteroids so that nearly the entire celestial northern hemisphere (roughly the same northern hemisphere we would see from Earth) is visible to both telescopes at all times. Since most of the mass of the solar system is in the plane (ok, most is in the sun, but the rest is in the plane), there should be little in your way. There are many asteroids in the main belt with high orbital inclinations, so you would have to account for this in the site selection phase, and perhaps make some efforts to move a few of them out of the way.
Now, a key to stabilization will be to rotate the asteroid. This will take some time and a lot of fuel, but by slowly rotating both asteroids at exactly the same speed, you will both improve stability of your optical platform, and provide a constant motion for each telescope relative to the other one. Again, this serves to restrict your field of view somewhat to one or other hemisphere.
Lastly, if you have enough money, you could mount separate telescopes on both sides of the asteroids, so that you can look at the northern and southern hemispheres at the same time with separate instruments. There are two ends of the axis of rotation, so you can be looking at both sides at once.
Multiple telescopes
If you want non-trivial frequencies with a non-trivial band, why not use a non-trivial number of telescopes? Since we're setting up shop on an asteroid of at least 1 km radius, and preferably more, there should be room for a variety of instruments.
The Hubble telescope has multiple instruments but only one mirror. Without going into specifics of what frequencies you are interested in, I think it is plausible to have two sets of instruments, one in the visual and near-infrared range, and another in the UV and/or X-ray range, each with their own optical mirror to focus on a variety of specialized instruments.
Position and timekeeping
The solution to your issues with combining the pictures from so far apart is to use high precision stationkeeping and timekeeping devices. For this purpose, a fleet of satellites similar to Earth's GPS system will do.
For example, a satellites could be set up in two orbits, one inside and one outside the Asteroid Belt. You will need enough satellites that at least two in each orbit are visible to each telescope observatory at all times. I believe that you will only need three in each orbit, but possibly four.
Using these satellites like GPS, if you are getting four signals at the same time, you can calculate your position accurately in fourspace (x, y, z, t). The principles of operation are the same as GPS satellites. These satellites already use atomic clocks and relativity adjustments for accuracy, so they will provide the location, direction, and timekeeping metadata to accompany each picture taken by your telescopes.
Post processing
With accurate enough 4-d orientation in spacetime, it becomes a relatively trivial matter to combine the pictures at a later time. The pictures and their metadata can all be beamed back to Earth for post-processing (the way that our deep space probes like New Horizons do now).
Conclusions
The only part of this which is not within our current technological capabilities is the heavy lift of dragging 100 m optical, IR, or X-ray mirrors 3 AU away to a suitable asteroid.
The 'image' combination technology is not much different from what LIGO is using for its disparately spaced detectors (Washington state and Louisiana); the only difference our orientation satellites need from GPS satellites is more power to push their signals over AU distances. And the telescopes in whatever bands you are interested in don't have to be any more powerful than the best we have on Earth (although, they do need to work in a vacuum, I suppose).
