I am sure I know the answer to this but I wanna make sure.
Ra has Flame Barrier in play. Baron has Backlash Field in play.
Ra deals enough damage to Flip Baron (he is still on his first side), Baron flips. Baron deals damage back to Ra. Ra takes the damage then deals damage back to Baron.
Did we play this right? Would Baron flip immediately or wait until the "exchange" was over?
You played it right.
I'm no expert but I believe you did it right. The moment Blade is reduced to 0 HP, before anything else happens, he would normally be destroyed (there might be exceptions on cards which do something when destroyed, such as Ambuscade's Sonic Mine or Unity's Bee Bot; it's unclear exactly how long those stay in play and whether they can resolve their effects from the trash, but other cards definitely stop instantly when destroyed, we have a long-standing ruling about Visionary's Wrest the Mind which makes this definite). Flipping him to his second side replaces his destruction, and you would then resolve both Backlash Field and Flame Barrier. Note however that if Ra reduced the Baron below 0 HP with the original attack, the extra damage does not carry over to the Baron; he always heals up to 30 HP after flipping. This Flame Barrier interaction might be one of the promptest ways to start cutting him down again, though.
Baron Blade (and indeed any action that happens on a trigger) would interrupt the 'exchange', BB will start on his flip side damaged by Ra. Nothing can halt the power of the flame!
(except Adaptive Plating Subroutines)
Thought we played it right but just wanting to make sure. it was a sad day because the baron still was victorious! :(