I just presume that such observation spacecraft makes sense to be made as a rather long term investment than some one-time fly-by thing like New Horizons, so the sensible thing to do is to have it in an orbit around the Sun at that (≥650AU) distance and to take multiple observations. Of course, the easiest solution may be for the orbit to be elliptical one, with the useful position only around its aphelion, in which case it will be able to observe only a single target indeed. But, I also presume that past the first few such elliptical orbit experiments the aim will shift to have the spacecrafts set on distant (and most likely circular) orbits that make possible observations at any time and thus its target would be something akin to a horizon rather than a single point, wouldn't it?