Congratulations on finding a method for baryogenesis that works! Indeed, it's true that if you have a bunch of black holes, then by random chance you'll get an imbalance. And this imbalance will remain even after the black holes evaporate, because the result of the evaporation doesn't depend on the overall baryon number that went into the black hole.
Black holes can break conservation laws like that. The only conservation laws they can't break are the ones where you can measure the conserved quantity from outside. For example, charge is still conserved because you can keep track of the charge of the black hole by measuring its electric field. In the Standard Model, baryon number has no such associated field.
Also, you need to assume that enough black holes form to make your mechanism work. In the standard models, this doesn't happen, despite the high temperatures. If you start with a standard Big Bang, the universe expands too fast for black holes to form.
However, in physics, finding a mechanism that solves a problem isn't the end -- it's the beginning. We aren't all sitting around scratching our heads for any mechanism to achieve baryogenesis. There are actually at least ten known, conceptually distinct ways to do it (including yours), fleshed out in hundreds of concrete models. The problem is that all of them require speculative new physics, additions to the core models that we have already experimentally verified. Nobody can declare that a specific one of these models is true, in the absence of any independent evidence.
It's kind of like we're all sitting around trying to find the six-digit password for a safe. If you walk by and say "well, obviously it could be 927583", without any further evidence, that's technically true. But you have not cracked the safe. The problem of baryogenesis isn't analogous to coming up with any six-digit number, that's easy. The problem is that we don't know which one is relevant, which mechanism actually exists in our universe.
What physicists investigating these questions actually do involves trying to link these models to things we can measure, or coming up with simple models that explain multiple puzzles at once. For example, one way to test a model with primordial black holes is to compute the amount heavy enough to live until the present day, in which case you can go looking for them. Or, if they were created by some new physics, you could look for that new physics. Yet another strand is to note that if enough primordial black holes still are around today, they could be the dark matter, so you could try to get both baryogenesis and dark matter right simultaneously. All of this involves a lot of reading, math, and simulation.
Best Answer
I don't think it's annihilation you have to worry about. It's my understanding that at typical energies adjacent (anti)baryons in a cluster confine their respective valence (anti)quarks too tightly for them to annihilate those in another (anti)baryon.
The real stability issue for a neutron/antiproton nucleus is that beta-decay of one or more neutrons to protons will reduce the electrostatic potential. Having said that, it's possible some specific numbers of neutrons & antiprotons are especially stable; you'd just need to deduce their equivalent of the whole of neutron-proton physics: magic numbers, drip lines etc. That's well beyond the scope of this question.
(A thought on terminology: if a cluster of nucleons leads to nuclear physics, we may as well call the antinucleon case antinuclear physics, while your mixed case would be sesquinuclear physics.)