On a different note, looking at this closely as we have it seems likely that there will be a rather terrified call from the cabin informing you of smoke in the cabin and a burning smell etc. This is to be expected until the reference items of the engine fire checklist have been completed past the point where the pack is turned off on the side with the burning engine.
Framer, you have a lot of funny ideas.
An engine fire is taking place somewhere inside the engine cowling (this is where the fire loops are located.. How would smoke get out of the cowl and in front of the engine (going at 180+ knots forward), then be sucked into the engine and through the compressor get into the bleed system?
What passengers will smell is a severe damage or engine surge/stall. Where the airflow is no longer normal in the engine, allowing exhaust to go back to the compressor and thereby going into the bleed system.
P.S.
The engine itself is always on fire as long as it is running. "Engine fire" is when there is a fire outside the engine.