Primordial black holes may be floating around, but they are very unlikely to be any to be closer to Earth than stellar black holes, given the kinematic evidence. (There could be faint hope for an isolated one of a fairly small handful within the Milky Way. FWIW, I don't think there are small PBHs. There might be gargantuan ones in the universe. There definitely isn't one of those hiding in our galaxy though, but maybe in our galaxy cluster near the centre of mass, wherever that is (millions of light years away, at least). There are some huge central black holes in some local galaxies that may have been seeded as primordial black holes -- the mechanism of growth of such ginormous black holes is not fully understood yet)). So they aren't really going to help you reach your thousand-year deadline, absent FTL travel.

Some researchers have proposed semi-seriously that there could be a small black hole lurking at the outer margins of our solar system. That obviously could be reached within a thousand years in principle (depends on how far out it is, although it must be considerably beyond Neptune's orbit), but we are perhaps decades out from a detection of it, if it even exists. The idea is, however, one of many reasons to build telescopes that are mainly to be aimed at objects in our star's immediate neighbourhood, to do very sensitive mapping of the metric the solar system sources (or, in Newtonian terms, our sun's gravitational field and the disturbances planets, asteroids, and other objects impose on it), and to do "nanolensing" observations on objects at a variety of distances. The Vera C. Rubin Observatory's Legacy Survey of Space and Time is a step in that direction, and will eliminate many possible hiding spaces for a nearby black hole.

There is an upper limit on proper acceleration proportional to mc^3/h-bar for massive subatomic particles like electrons, from Caianiello (1984) <https://doi.org/10.1007/BF02748378> (free PDF <https://link.springer.com/content/pdf/10.1007/BF02748378.pdf>) arising from uncertainty relations. We have no hope of producing that kind of acceleration as even with the best possible superconductors an Earth-bound accelerator can manage much less than a millionth of that <https://arxiv.org/abs/quant-ph/0407115>, and we need to be much closer in order to make a black hole out of small charged particles. Large particles are even harder to accelerate extremely. We also have a power-density problem: what fuels such an accelerator? Alternatively, accelerating less extremely means much longer linear accelerators, or nearly equivalently, less curvature in a ring accelerator. I think we're looking at a ring about the diameter of the orbit of Jupiter, or a linear accelerator crossing a large chunk of the inner solar system, but that's just a guess (someone has probably worked out some comparable size for ultrarelativistic heavy ions (lead or gold) though).

So the tl;dr is that it's not impossible, and it's easier than trying to throw large stars at one another, but neither is really plausible within your 1000 years. And neither is easier than sending a probe to V616 Mon. and waiting several thousand years. And we might do better by making really good observatories in and around Earth. Technically we could do all three, I guess, but we're more likely to live long enough to see improvements on the results from the <https://eventhorizontelescope.org>.

And of course, what I said about your three-digit timeframe goes double (if you pardon the expression) for your two-digit one.