dg.differential geometry - Local Triviality of an Associated Bundle

@Scott: I'm going to add this as an answer because it a bit long for a comment: Sorry, but I'm a bit slow and want to be certain that I understand exactly what you're saying: If $(p,v) in P times_{rho} mathbb{C}^n$, then $$(alpha_u times id)(p,v) = (pi(p),g_p,v) in U times G times mathbb{C}^n,$$ for some $g_p in G$, depending on the choice of $alpha_U$. This is not in $U times mathbb{C}^n$ as we would want, moreover it's not even well defined. So by $alpha_U times_{rho} id$ do you mean the composition
$$
(id times m) circ (id times rho times id)circ (alpha_U times id),
$$
that maps
$$
(p,v) mapsto (pi(p), rho(g_p)v)?
$$