Thanks to everyone for the advice and information. There are a few non-US airlines that allow for outport base stationing (and commuting) in the US. For now my husband is actively pursuing that direction.
Most of what everyone is saying is correct; I have also tried to confirm this information through my extensive research for the past few days. It looks like my husband will have to convert his licenses through ATP-CTP flight school training before even considering applying to any airline above the regionals. We are also pursuing extending the timeframe for when we would need to permanently relocate to the United States (to keep his permanent residence visa valid while abroad). This will give him more time to convert his licenses and aggressively pursue the majors, if, of course, they consider his application.