As I understand it, rather than "finite definition of a model for reality" it's really more a matter of having a single model. No one model can cover everything but multiple models can, right?

What would be the difference between multiple models and one model containing all of them? You can keep adding on to the model, or mixing and matching models, and you will never get one that immune to go goedel. Either you have model of physics with contradictions, or you have one where there exist questions that cannot be proven true or false, in other words, undecidable.

Right. I can't imagine a single model for reality. That is, in the sense that you start the model running and it reproduces all events everywhere from the Big Bang to whatever happens eventually.

But models that generate results that can be tested, through observation and experiment? Sure. Maybe even arbitrarily integrated models.

However, there's uncertainty throughout. So models can't be deterministic. Certainly at the "ends", at the quantum level, and at the level of consciousness.

A model doesn't have to be deterministic, use finite quantities, or even be possible to execute in a simulation. Math often does not concern itself with such things. Undecidability is concerned with the set of rules themselves. Even if it is impossible to actually simulate, is there a finite set of physical laws which drive all things? If yes, and if you can construct the basic requirements for counting and making logical statements and such within that system, then you are subject to incompleteness.

That said, it's interesting to think about a universe with truly infinite rules. Each physical law could have minor exceptions caused by smaller more detailed phenomenon. Each time you would discover some new principle, it would reveal more yet unknown questions, a fractal of infinite knowledge to be refined and science to do. But I think most scientists hope for a finite set of underlying rules for reality.