Could you maybe provide me with a more precise pointer to where your last paragraph is in the reference you mention?

@TimPerutz Thank you for your answer. Unfortunately I'm not knowledgeable enough to follow all the details, but I found a reference that makes a similar statement. Although only for simply connected compact groups (Dijkgraaf, Edward Witten: Topological Gauge Theories and Group Cohomology). Just to be sure: The statement you provided is also true for non-compact simply connected groups, right?

@TimPerutz Do you maybe have a reference where it is stated that G- bundles over oriented 3-manifolds with simply connected groups are trival?