Hey, a gravity tractor beam! Supraradiant scattering?
PBS Space Time very recently released a video about this topic, which is far better than any explanation we ourself could give you.
Unfortunately, frame dragging makes it harder to add angular momentum since in order to do that the beam would have to be faster than the ergosphere it's exchanging with. If it can be shown that there are paths along which the beam can slow down and still escape, exchanging angular momentum in the desired direction, then the limit is prolly set by how fast the beam can be.
Personally, we'd be surprised if it turns out the kinetic energy which comes with the added angular momentum is not enough to keep the outer horizon ahead of the inner one, at least for macroscopic black holes. It's more than a little suspicious that charge accumulation works that way, with electrical potential being enough to keep it from being superextremal. We'd find such surprise rather delightful, though!
At the very least, we could use it as a flywheel energy storage scheme. If we're willing to wait gobs and gobs for it to evaporate, and keep it spinning right fast close to the end of its evaporation... (frame dragging radiates angular momentum away)
Schwarzschild radius is still a linear factor of mass, so it makes sense to ask whether a superextremal black hole is possible at some small mass. An internet search leads to "black hole electron hypothesis," so apparently the answer is that it would look suspiciously like an electron? On that size scale, the horizon dimensions would be hilariously small compared to quantum fuzziness, so it's not really meaningful to say the body's mass is "located" inside that horizon. We guess if you strip a singularity of its relativistic horizons, quantum mechanics takes over to preserve its modesty anyway?