How would we know the difference between an expanding universe and a collapsing one?

[Sarkus]

If that were the case, then everything in the universe would be moving inexorably towards the centre of the hole. Instead, we see the universe expanding in all directions.

That's only if you think classically about things inside the event horizon, which black hole cosmology certainly does not.

If a black hole has constant mass, the spacetime around it is essentially static. It does not "stretch". Unless I'm misunderstanding what you mean, perhaps. What do you actually mean?

You have to think about two realms of spacetime, one in the parent universe and then our own. Or one outside the black hole and one inside. Outside the black hole things behave classically. And if things behave classically inside a black hole then they'd behave as you suggest, e.g. matter falling into the singularity etc. But that's not what the proposal is. I'm not entirely sure what the various proposals suggest, even the simplest of them, and it's not exactly the most intuitive idea, and is (almost?) unscientific - in that I'm nor sure how it can be falsified, or even tested. But it's out there, and there's increasing buzz about it at the moment. Maybe just slow-news-days in the scientific community.

Why would space stretch inside a black hole? What would cause the stretching? You'd need to add or subtract mass, wouldn't you?

You're thinking classically, rather than thinking about the way things are proposed to behave. Classically, you're right, but the hypotheses explain away such issues. They're worth a cursory read about, even if just as an interesting idea rather than anything which will shatter your worldview, or affect you practically.

Black holes all have finite mass. Infinite universes, on the other hand, would have infinite mass.

Who says the universe is infinite? Certainly the observable universe has finite mass, but we can only say that the total universe may be infinite. We just don't know.
Black hole cosmology leans heavily on the finite-mass universe, but there are versions of the idea, I think, that do allow for an infinite-mass universe. From what I understand of things, though, if the universe is infinite (and I think that is the most widely accepted view - but that doesn't make it fact) then it would put a fairly large hole (no pun intended) at least in the more basic/simple black hole cosmology ideas.

Imagination is all well and good, but at some point the rubber has to hit the road. Is there any physical theory that you're aware of which suggests that what you imagine could actually be the case? Or are you just tossing the idea out there to see if somebody else can make some sense of it?

There's no direct evidence, and at the moment it is barely up to the bar of being scientific. But it is based on physics, and there are some interesting maths, apparently, that suggests it. For example, for the given mass of the observable universe we would expect a black hole to be about the size of our observable universe. i.e. if you took all the matter and squashed it into a singularity, the event horizon would be roughly the same size as our universe. Could just be coincidence.

But the basic idea is speculation at best at the moment. Not even science. Not testable, not falsifiable. Although it may one day become such, at least to those who actually (think they) know what they're on about. But it's an interesting idea.