Tubular neighborhood: compact support for the pullback of a form with compact supoprt

As I have pointed out here:

Projection formula, Bott and Tu

The integral projection formula given in Bott and Tu is incorrect. This is used later in the book, on page 67 to prove that Poincare dual of a closed oriented submanifold in an oriented manifold and the Thom class of the normal bundle of the submanifold can be represented by the same forms. The proof is on page 67 here:

http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf

As was pointed out in the answer by Eric Wofsey, the solution would be for the pullback to be a proper map. This is not available. But, since the proof uses a tubular neighborhood which can "shrink", could we use the shrinkage to find a compact form approximating the pullback with arbitrary accuracy? Or any other way of going around the error in Bott and Tu?

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.)
• The Thom class is defined in terms of the Thom isomorphism, and the Thom isomorphism only exists if you make this extra assumption (see my edit). Alternatively, the explicit construction of the Thom class via the $e_*$ map defined on page 38 (as alluded to on page 63) manifestly gives a form whose support is proper over the base, since its support is restricted to a fixed compact subset of the fiber (the support of $e$). This will continue to be true when you patch Thom classes together with a partition of unity in the general case. – Eric Wofsey Nov 23 '16 at 19:51