You don't just have to get the probe there. (And which probe? You have to design and build it first.) You have to get it there, have it get data, and have the data come back. Otherwise you've just thrown a rock, which, yeah, if we spent a fortune we could certainly throw a rock out vast distances very quickly.
So, you either:
Do a flyby, which requires a slower speed. New Horizon's speed was good for gathering data from Pluto and sending it all back. The hypothesized PBH is small enough to fit in your duffle and probably much darker.
Enter orbit. Easy enough with a 5-10M object. But the orbit has to be a useful orbit. So either you burn a tremendous amount of delta-v to reduce your tremendous speed so you get a compact-enough orbit; which requires even more energy to get out there, or you craft the orbit such that the object itself helps slow you: but that takes time and more speed means more time.
Studying it might allow us to finally unify GR and QM.
1. Humans aren't going to survive in space for 10 years. It's questionable that they'd even survive a trip to Mars without getting riddled with cancer from the cosmic radiation. Sure, if you built a big enough ship to provide some really effective shielding, it's technically possible, but that ship would be enormous and far beyond our current capabilities. I don't think it's feasible at all to launch such a ship from Earth; it would need to be assembled in space.
2. Project Orion is just an idea on paper; it's not within current technology, because no one ever built it. We don't "have the technology" at all. We don't even have the technology to land humans on the Moon. We did decades ago, but we no longer do: all the people who knew how to do that are retired or dead, so we'd have to start over. Of course, we can build powerful rocket motors easier now since we do so regularly now, so building equivalent Moon-landing capability is no longer as difficult as in the 60s, but a lot of things would have to be partially re-invented (e.g., the lander itself, the rover, etc).
3. Does your time estimate include the time needed to decelerate, so the ship doesn't just zip by the black hole with barely any time to collect data? (And if there's people on this ship, they might want to return to Earth...)