The balloon model of an expanding universe is of course an analogy, useful to explain how galaxies can move apart without there being a central point they are all moving away from. One should recognize that it is not necessarily a good analogy when we start changing the assumptions a lot. It is even a bit of a problem when thinking about flat and negatively curved universes, since balloons have positive curvature.
Now, what happens if you collect a large amount of matter from the currently nearly static distribution into one region? If you just place it uniformly in a spherical region the local curvature will still behave according to the Friedman equation and you get the same spherical symmetry as originally, but with a different radius of curvature and expansion rate. So the balloon gets a part that acts as a smaller, less rapidly expanding balloon (don't think about how that connects to the rest of the balloon: the analogy to a physical balloon breaks here, despite the actual transition being fairly mathematically OK). If there is enough matter you get Einstein's static universe, a non-expanding but curved part.
Generally, in GR people often do things like this mathematically: cut and glue different solutions to each other, taking care to match curvature at the edges.
If you add more it will start contracting. Note that a big (so that central parts cannot be causally affected by the outside) even distribution in principle would make a little big crunch collapse rather than a black hole, but in practice that is not going to happen since you cannot move matter outside its own lightcone to overload such a big region. And if you just pile up matter in one point you get a black hole.
Black holes do not fit the balloon model well. There is Flamm's paraboloid, a 3D embedding that has the same curvature as the space, so you could imagine replacing a part of the balloon with it. Now the balloon has a hole... a hole which does not leak any air, and whose shape stays the same despite an expanding outside balloon. The analogy is breaking again.
But there is work done on how single huge black holes in an otherwise empty universe work, and they get pretty weird for very massive holes.