I would like to just comment but don't see how. Christian, here's an answer to your last question.
For based spaces, you can just do what you want by hand, as Fernando suggested, taking care to use
disjoint basepoints to make your attaching maps of $A$ based. You are right to complain that the
interplay of $h$ and $q$-model structures is not obvious. Based spaces are of course spaces over
and under a point. In ``Parametrized homotopy theory'', Sigurdsson and I generalize to parametrized spaces, which are spaces over and under a give space, and then the combination of $h$, $q$, and related model structures is surprisingly delicate. In that book, the answer to your original question is axiomatized in a general model categorical context in 5.4.1 (see (v)) and the axioms are verified for parametrized spaces in 5.4.9. But that is like hitting a thumb tack with a sledge hammer. Maybe
it will help to add that in the direct argument you do need to know that a wedge of weak equivalences is a weak equivalence, and that uses the well-pointed hypothesis.