Is it permissible to pay money to an intermediary to obtain residency in a disbelieving country, knowing that returning to Muslim countries is very difficult and requires my niqabi wife to remove her hijab for fingerprinting, and that not obtaining residency leads to extreme hardship in living and working, and knowing that my intention is to return to Muslim countries after arranging my business affairs?

It is permissible for a wife to uncover her face or hands when necessary for departure procedures from non-Muslim countries. However, this is not an excuse to remain in them. The issue of sustenance and fear of disappointment does not permit committing forbidden acts, for sustenance is decreed by Allah. Bribery is forbidden and is only permissible for obtaining an indispensable right out of necessity, with the sin falling upon the receiver of the bribe.