How exactly would you distinguish between 'proven' fundamental and merely 'considered' fundamental?

I mean, sure, some laws like Heisenberg's Uncertainty Principle can be proven mathematically from the axioms of a theory, but how exactly would that make it more fundamental than a law that is considered an axiom itself?

I'm not too knowledgeable in these things, but I'd guess it'd really just depend on the same things everything else does in science: supporting data.

Certain things have an immense amount of supporting data.

The famous philosopher of science Karl Popper disagrees, and few disagree with him - science can definitively disprove, but not definitively prove anything.

PS FWIW Heisenberg's Uncertainty Principle isn't understood now as it was then - that principle as it was has been overturned.

How do you ever prove something in an empirical science?

There are various levels of how rigorous evidence is. Apparently, the law in question was made based on assumptions about the materials involved, and as someone else pointed out there was a ten year gap between the mathematical explanation and actual implementation, so I guess these materials are quite complex.

Also, this does not invalidate the previous law necessarily; we still use Newton's laws because its assumptions hold under most circumstances.

By building things that work.