Godel's proof is about our inability to prove some theorems mathematically, but that does not mean we cannot prove them scientifically. Such proofs, of course, will suffer from the same problem all scientific proofs have - a certain probability that even though our model is wrong, somehow by pure chance our tests ended up showing otherwise (in technical term - non-zero p values)

except that part about not trying to prove the parts that are crucial for the scientific method… You are just wrong about that

I'm not saying that one must never attempt to prove these foundations. What I'm saying is that if you try to prove them empirically (as oppose to how they are usually proved - mathematically) using the scientific method, you will run into the circular reasoning fallacy:

     -----> foundations >------
    /                          \
 proves                      proves
    \                          /
     --< scientific method <---