Noob Statement to follow..... Wouldn't starving the fire of oxygen through a steep nose dive or cutting the engine (and/or flow of fuel), BE a fire suppression technique? I get that we don't have an "Engine 2 fire extinguisher button"...
Your dive might just intensify the fire causing structural damage before reaching a point where you "blow" the fire out.
With gunfire damage many times the fire is oil leaking from a cracked engine case or gunfire damage to fuel or oil lines. On a B-17 or B-24 each engine has a 32 gallon oil tank mounted next to it...32 gallons of burning oil will do a lot of damage and there is no way to shut the flow off. Oil burns very hot.
Inflight fires scare the poop out of me, and I'm almost fearless.