The military only has a small number of pilots trained on the C-40, and they're needed for actual military operations. Even with basic type familiarity, every airline has it's own unique operating procedures and so new pilots always have to pass an extensive training program including classes, simulator time, and rides with check pilots. This is mandatory for safety. No legitimate airline would take shortcuts there just because of a temporary crew shortage. If a Southwest flight crashed because the Captain and First Officer weren't following identical checklist procedures in an emergency that would be far worse than cancelling a bunch of flights.

A fairly high percentage of airline pilots are also part time pilots in the National Guard or Reserves, so they may be simultaneously rated for a certain airliner type as well as a military aircraft. But that doesn't really help for this situation.