If you're changing the dash anyway why not start with that. Make sure the heater core and evaporator are good, all the doors/functions work, can't say if they're mechanical, vacuum or a combination of both. Then at least, you'll get air flow through there then add the Compressor/condenser later?
Not to be a downer but a 22inch, no A/C should be fine. A shroud might help but you may have a restricted flow in the radiator, looking at a re-core? Just my opinion.
If my experience with a 68 Fury, (No A/C), applies. It's a chore but pulling the whole dash first would make it all easier. Saving that, a tip I got on this site was to unbolt the right side fasteners and pull out the dash to help with the glove box, (even if you get a new box liner you'll want...
Not sure what the 360 is, stock or built, might want to look at restrictions in the radiator, water pump or block. Any over the counter hoses, (NAPA etc), should perform fine and not bulge like that unless there's high pressure.