Thanks Ben. I'm not sure I understood the last bit, I understand that there is a 1-1 correspondence between sections of the vector bundle and equivariant functions on $P$. But here we are not making things descend to $M$ but to pullback bundles from $M$ to $P$. After a long though I came to the conclusion that we need the $G$-invariant vector fields because the requirement of the $G$-invariant splitting implies that the vector fields of $TP$ must be $G$-invariant (this follows directly from what I wrote above $gv_p = X_{pg} + H_{pg} = v_{pg}$). Please let me know if you agree with this