You can do what ever you want if it is justified in the event of an emergency. An engine fire is an emergency an engine failure is slightly less so.
I don't reckon you can talk your way out of deviating fron the SID by saying "Our company told us to!"
If you are unable to follow the SID you should be following the published contingency plan or on a radar vector once requested. The radar vector plan kills two birds with one stone as it puts the terrain clearance in someone elses hands whilst alerting others of your plight.
The aircraft should be on autopilot once above acceleration altitude and trimmed and that reduces workload considerably.
Just the checklists to read now. No big deal, was it?