This requires sending something to at least about 548 AU and then effect falls off from there but anything you send that far will be going at a velocity that would keep it going. You would be in effective range for some time but basically you'd need to keep sending satellites to that distance in order to keep using the technique. You'd also want to send them into different directions in order to image different parts of the sky.
Couldn't you do something like a Hohmann transfer orbit ? As I understand it, adjusting the orbits far out there isn't as demanding as one might think (you don't have to spend the same energy you took to get out there to get into an orbit. Of course, this also means you will not be able to stay in one place and continuously map the same area, but you can map all the space within a disc?
It takes decades however to finish Hohmann transfers that far out apparently.
Rabbithole: Some calculations that may be entirely wrong would suggest that a <10000lbs spacecraft might be held "in place" for 100 yrs with a small fission reactor and an indium-based FEEP drive with about 15lbs propellant.
If you circularize the orbit at 550 AU (which is costly but possible, at ~1km/s of delta-v), then you end up with an orbital period of >10000 years (to complete your whole-sky survey in one plane).
But first-- the price to get that circular orbit is steep, you end up imaging empty space for a lot of the time (no real way to "skip ahead") and your orbital speed vector makes you leave the observation cone for any interesting target pretty quickly.
In conclusion: You probably really want to stick with "one telescope probe per target system", and to give them one-shot orbits because those are cheap, reasonably fast and inside the observation cone (for your single target) for longer.
It’s basically imaging one exoplanet per satellite. Would be good to figure out a way to slow down at 500 AU. In order to maximize the time for observation.
It would be most likely a whole solar system. Consider that new horizons only had hours during its flyby to collect all the telemetry that it did. Imagine having 600 pixels of resolution for an exo planet for up to 15 years. Plus all the other telemetry you'll be collecting about the galactic space near us.
With a Jupiter/solar oberth boost we can at best do around 20 AU per year, so best case scenario it would take atleast 25 years to get there, which honestly isn’t bad at all. For comparison voyager does around 3.6 AU/year.
A series of permanent stations near the vicinity of the oberth burns at Jupiter and the sun could add a lot of Delta V by firing a laser at the probe and the probe could fire one back. Even more so if it was a series of laser boosters strategically placed around the solar system. The numbers start to get really crazy.
Nuclear-powered ion thrusters could solve this issue. They provide low acceleration for a long time consuming very little consumable. This would allow the telescope to stay at the right position for observation.