1. Space is big.
2. Space is REALLY big!
3. This thing would be tiny.
4. This thing would be invisible.
Combine these four together and you'll quickly realize just how difficult even scoping out the probe's requirements and such would be. Would be cool though.
"Searching for a Black Hole in the Outer Solar System", https://arxiv.org/abs/2004.14192
"And one would like to launch hundreds of spacecraft (at least) in different directions so that some would come within dozens of AU of Planet 9, rather than hundreds of AU."
Their example of a neat-but-good-luck project could actually achieve the goal is [1]. The spec requires ground lasers that's powered by a 1GW nuclear power plant for propulsion. And, "According to The Economist, at least a dozen off-the-shelf technologies will need to improve by orders of magnitude." [2]
[1] https://breakthroughinitiatives.org/initiative/3 [2] https://en.wikipedia.org/wiki/Breakthrough_Starshot#Technica...
If you can get a spacecraft somewhere in the vicinity of a black hole of decent mass (like a 10km asteroid equivalent) we could probably detect changes in trajectory of the spacecraft from the gravitational attraction.
Or, here's an idea, blow out a tonne of radar chaff in the vicinity of the black hole and watch for how it disperses via radar.
We could also try detecting Hawking radiation that a BH should generate, though that might be pretty faint (I have not done the math).
You'd have to look for a very precise spike in the EM spectrum at the electron-position annihilation energy equivalent.
Despite being raised on space opera and hard-sf and having a science degree, I am pretty bearish on most space exploration. Having said that, a nearby black hole wouldn't merely be nifty. For reasons we do not understand, the event horizon stops honoring conservation of baryon number (and lepton number, and other numbers). I consider this important because replication of that would allow for direct conversion of uncharged mass to energy, just as any neutral particle crossing the event horizon has its energy (eventually) returned to us. If Hawking radiation is true (we have yet to experimentally verify it), then we know that mechanism exists. Can we do it without a black hole?