I understand that I'd need a type rating on a recognized multi-engine type to unfreeze an ATPL
Be careful here. It's not a TR on a multi engine type that you need. It's a TR on a multi
PILOT type. Sorry if that sounds pedantic, but as stated above, there are many multi engine aircraft that are classed as single pilot (the King Air, as was pointed out) and would not therefore unfreeze your ATPL.
As far as the Mil Bridging goes, Beagle is much better qualified to advise you there!