John, your question is an advertisement for Johann Sigurdsson's thesis and our book ``Parametrized homotopy theory'', http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf, which is where the results of his thesis appear. Section 6.1 there explains many of the pitfalls of the obvious model structure that you start with in your question, which we call the q-model structure. Following James, we call your retractive spaces "ex-spaces''. Johann's thesis gives a new model structure which we call the $qf$-model structure, starting with a $qf$-model structure on spaces over the given base space $X$, rather than the $q$-model structure on spaces, and going from there to ex-spaces. The rest of Chapter 6, pages 100-108, is devoted to proofs and explanations of that model structure and its properties. That is probably too lengthy to summarize here. The model structure does have the same weak equivalences, and it is cofibrantly generated. As you expect, the subtlety is in the precise definition of the generating cells, and that is really subtle and entirely due to Johann. Of course, you must restrict to quite special generating cells, and this does raise problems, forcing us to introduce some variant model structures in Chapter 7. The identity functor is a Quillen equivalence from the $qf$-model structure to the $q$-model structure, so the homotopy category is what you want. Incidentally, we generalize to the equivariant context in Chapter 7.
Edit: In light of the comments, here are my grounds for skepticism, from first principles.
Given an adjunction $(L,R)$ between categories $\mathcal C$ and $\mathcal D$ and a cofibrantly
generated model structure on $\mathcal C$ with generating sets of cofibrations and acyclic
cofibrations $I$ and $J$, the natural way to try to construct a model structure on $\mathcal D$ is to let $R$ create the weak equivalences and fibrations and to take $LI$ and $LJ$ as generating sets. Formally, the maps that satisfy the RLP wrt $LI$ are then the acyclic maps that satisfy the RLP wrt $LJ$. I know of no examples where the same conclusion holds defining acyclicity in terms of a different class of weak equivalences. The relevance is that (modulo basepoint details) John starts with the adjunction $(r^*,Sec(X,-))$ between ex-spaces over $X$ and based spaces. His proposed fibrations are created by $Sec(X,-)$, and his proposed generating sets are $r^*I$ and $r^*J$. However, his proposed weak equivalences are not the weak equivalences created by $Sec(X,-)$ but rather the maps that are weak equivalences on total spaces.