Twan, thanks for the detailed comments!

Program A needs F(N), not on N itself. So it either needs to represent F in addition to N, or just F(N). Since F is computable and therefore has a finite program, you can just include the whole thing.

Yes, the intended interpretation is that A represents F too, as do all statements that mention F. That's why in the bounded Lob's theorem the function L is stated to be computable.

It is not at all evident that the length of the proof grows linearly or even polynomially with the length of the program A. This depends on the theory T, it might have a very inefficient way of denoting proofs.

That's true. I guess I'll have to include that as an explicit assumption somehow.

I am not sure if the corresponding variant of the bounded Löb's theorem holds, though.

I think it doesn't hold. Actually it's a nice exercise for everyone to try coming up with alternate formulations of the bounded Lob's theorem, like switching around N and L(N), and see if they can be made to work. It's surprisingly tricky.

The formal theory will need some additional symbols to tie everything together, so the total proof would need 2*N+K symbols, where hopefully K doesn't depend on N.

That's true. Maybe it's easier and more general to just say the length of the total proof is a computable function of N, that way everything seems to work fine.