Right now, with the current engineering capabilities of humanity, we could get a probe to relativistic speeds fairly easily (in the sense of how many fundamental engineering problems would we need to solve).

If economics aren't a factor, i.e. humanity decides that its collective goal is to make a probe go brrrr as fast as possible towards a black hole, and everyone is working towards that goal (money is no object) then it's actually not that hard. We can send a probe up with a small mass and a huge surface area light-sail, then build high-power laser arrays all over the earth en-masse to point at it.

I haven't done the math, but you could get something up to at least 10% the speed of light this way, probably even 50%.

The bigger problem is a) if we want to send something that has enough mass to actually contain the necessary functionality to transmit back to us from that far, then it becomes much harder to achieve any relativistic speed, and b) it will probably destroy itself after colliding with a dust particle.

And of course thats on top of the fact we'd need to figure out how to power it, we wouldn't see results for 4500 years minimum, and the second it hits that event horizon its gone from our reality forever anyway.