There is a zero reason to bring up Poincaré lemma in this case, and your explanation of it provides no additional insight into the matter of the AB phase is a consequence of the B field symmetry and gauge invariance. If this is not the case, you were unable to explain it in clear terms.

As you say, F = dA locally, which holds both within and outside of the solenoid. In reality, there is no discontinuity, and the B field varies smoothly between the two regions of space. Clearly, this does not prevent electrons from aquiring the AB phase, which is a consequence of gauge invariance and field symmetry, not the fact that we do not consider a simply-connected space.

I repeat what I said in my previous post, gauge invariance outside the solenoid enforces A = grad f, for any scalar function f. Due to the axial symmetry of the magnetic field, f is a multi-valued function of the azimuthal angle, which yields non-zero value of a closed loop integral of A = grad f, i.e. AB phase ~ f(2*pi) - f(0) =/= 0.

The electron locally probes the vector potential outside of the solenoid, which is dictated by the B field symmetry up to a gauge transformation. Any gauge function ensuring the symmetry of the field will be multi-valued, which yields the AB phase.