You do not have beliefs about the FTA; you have opinions on the usefulness of the definitions which imply it.
This is false as a psychological description of my personal state of mind. I don't know the precise definitions that entail the FTA and I certainly don't know a proof. (In particular, I don't think I could give you a correct construction or definition for the real numbers.) I believe in the theorem because I've seen it asserted in trustworthy reference works. Somebody somewhere might have beliefs about the theorem that were tied to their beliefs in the definitions, but this doesn't describe me. I can believe the [deductive] consequences of a claim without knowing the definitions or being able to reproduce the deduction.
Here's a related example with a larger bullet for you to chew on. Suppose I have a (small) computer program that takes arbitrary-sized inputs. I might believe that will work correctly on all possible inputs. Is that a belief or not? It can be made as rigorously provably correct as the FTA.
When I say "the program is correct", I am not saying "it is useful to construe the C language and the program code in such a way that...". I'm making an assertion about how the program would behave under all possible inputs.
Beliefs about computer programs might feel more empirical than beliefs about theorems, but they are logically equivalent, so either both or neither are beliefs, it seems.
Please observe that one of the possible inputs to your computer is "A cosmic ray flips a bit and turns JMP into NOP, causing data to be executed as though it were code". In other words, your proof of correctness relies on assumptions about what happens in the physical computer. Those assumptions are testable beliefs, just like the intuitions that go into geometry or the FTA.
Very brief recap: The logical positivists said "All truths are experimentally testable". Their critics responded: "If that's true, how did you experimentally test it? And if it's not true, who cares?" Which is a fair criticism. Logical positivism pretty much collapsed as a philosophical position. But it seems to me that a very slight rephrasing might have saved it: "All _beliefs_ are experimentally testable". For if the critic makes the same adjustment, asking "Is that a belief, and if so -" you can interrupt him and say, "No, that's not a belief, that's a definition of what it means to say 'I believe X'."
A definition is not true or false, it is useful or not useful. Why is this definition useful? Because it allows us to distinguish between two classes of declarative statements; the ones that are actual beliefs, and the ones that have the grammatical form of beliefs but are empty of meaningful belief-content.
It seems to me, then, that both the positivists and their critics fell into the trap of confusing 'belief' and 'truth', and that carefully making this distinction might have saved positivism from considerable undeserved mockery.