In a local universe empty of any matter except a Kerr black hole and an observer, that observer is spinning at the same rate as the black hole and observes it from a great distance directly above its north pole. Will he deem it to be a Schwarzschild black hole?
If the observer is spinning at a different rate, will he see a Kerr black hole with different angular momentum?
No. A coordinate transform cannot change the angular momentum (or mass) of a Kerr black hole. In particular, the Schwarzschild black hole is spherically symmetric. The Kerr black hole is not.
By the way, observers and coordinate systems are independent concepts. The "frame of reference" language tends to suggest some kind of connection between them, but they are independent, even though they are conflated with each other in many introductions to special relativity (where we tend to consider only a very limited class of coordinate transformations). In any case, whichever of these concepts the question might mean by "frame of reference," the answer is still no: we cannot make a rotating black hole look like a nonrotating black hole just by changing the frame of reference.
You can't really evaluate how a black hole in our universe works by comparing it to a thought experiment of a universe containing just the black hole. There is a phenomenon called "frame dragging" in which the rotating black hole alters the spacetime around it through its rotation. However, in the reference frame of an observer rotating with the black hole, the rest of the universe is rotating, and from this observer's perspective, its rotation causes frame dragging in the opposite direction. The effect of the rest of the universe is that the black hole's frame dragging is partially cancelled out (and the net frame dragging decreases as one goes away from the black hole). So while the Mach principle does assert that a universe containing nothing but a rotating black hole would be indistinguishable from a universe containing nothing but a non-rotating black hole, that doesn't mean that a rotating black hole with our surrounding universe is indistinguishable from a non-rotating black hole with our surrounding universe.
Relativity says that the rules for what things do don't depend on the reference frame, but doesn't mean that reference frames can't differ in how things are affected by the rest of the universe. A car traveling at the same velocity as the Earth will experience different phenomena than a car traveling at a different velocity, but that's not because there are different physics operating on them, it's because the surrounding matter is interacting with them differently.
