There is nothing wrong here. The projection formula is perfectly correct as long as you change the definition of $\Omega^*_{cv}$ (to require the support to be proper over the base). That is, you should consider the definition of $\Omega^*_{cv}$ to be changed in this way throughout the entire text. The Thom class $\Phi$ to which the projection formula is applied on page 67 is by definition an element of $H^{n-k}_{cv}(T)$ and so, assuming you have corrected the definition of $\Omega^*_{cv}$ from earlier, the projection formula does apply to it.
In fact, more strongly, you need to change the definition of $\Omega^*_{cv}$ or else you can't even define the Thom isomorphism at all. The reason is that integration along the fiber isn't even well-defined if your forms don't have compact vertical support in the stronger sense. You can integrate a form pointwise along the fiber, but the result may not be a smooth (or even continuous) differential form on the base (the counterexample I gave in the other question works again here).
(Of course, you have to check that everything else in the text works with this changed definition, but this should be straightforward. The only step that actually involves constructing any specific forms that you might have to worry about having proper support is in the (omitted) proof of Proposition 6.16. In that proof, an inverse to $\pi_*$ is constructed using a map $e_*$ as on page 38, and this $e_*$ does give forms which have proper support because the support on each fiber is contained in the support of $e$, which is a fixed compact set.)
