Black holes can orbit each other (we know this from the gravitational waves from their eventual mergers) and there is no particular reason against a small black hole passing through the ergosphere of a large black hole.
The main problem is avoiding a merger. The the two-black hole spacetime has no analytic solution and numeric simulations are messy, but a merger will definitely happen if the event horizons overlap. In any case, a rotating black hole has an ergosphere with an equatorial radius of $2GM/c^2$ and event horizon somewhere between this and the limiting radius $GM/c^2$. If we use the extremal case, then we can presumably just barely fit in nearly an identical mass and rotation black hole in the ergosphere. More likely a much smaller black hole could fit in. The direction of rotation would matter for the dynamics, since counter-rotating black holes would tend to coalesce (see this page for a plot of the relevant radii and where the innermost stable circular orbits are).
A single black hole passing through the ergoregion would not be able to perform the Penrose process, but we can imagine a binary black hole pair that is ripped apart by tidal forces and then presumably the remaining hole would be ejected with a lot of energy. So, yes, we could get fast small black holes this way.
In practice this system would lose a lot of energy through gravitational waves, making it likely that the holes would coalesce anyway. The exact limits require numerical relativity simulations to estimate, and this one would be a tough one.
(Note that the gravitational waves also carry away energy, up to 1/8 of the total mass of two equal-sized holes merging, but this energy is very hard to capture. Penrose did note in his classic paper that Misner was first in proposing black hole mining using a variant of this.)