I may have missed it but are there helos out there that DON't show which engine has the fire: T handles / associated fire wires/ lights?
This being the case (ie: ambiguity regarding which engine) I can understand a height climb whilst identifying the cause/source. BUT in helos which immediately indicate which engine is "on fire" why would any sane pilot climb away from the biggest fire extinguisher in the world?
Given that the a/c is designed to advertise which engine is affected, I would come off the rig at Vy and cruise at 50'/100' (day/night) whilst stabilising the fire. IF it couldnt be contained it would mean a very short trip into wind and a controlled ditching. Fires do give pilots a little time to sort their lives out, but a real (uncontained) one "probably" wouldn't allow you to get away with a climb to 500' and back again to sea level.
Something that occasionally goes unnoticed is a post shut down fire where T4/PTIT/outlet temp is rising. This is an internal fire and cannot normally be extinguished by onboard fire suppression systems as these are directed at the fire bay (external). The only way to deal with this is engine starter blow thru and hope it extinguishes the fire - or else you're back to uncontained fire.