Our company SOPs mandate a straight out departure or follow an emergency turn procedure (we call this SEOP)if that is published, but not the SID.
Take-off performance (i.e. the Max allowable TOW you can depart with at a given rwy and conditions) is based on a predetermined obstacle clearance according to FARs/JARs. Obviously all airlines want to uplift the max possible weight to take advantage of max payload/range missions, so all Emer Turn procedures are based on the track that requires the minimum climb gradient.
ATC units are supposedly notified of your company's emer procedure, but I wouldn't bank on this. Thus, if you face an engine fail there's yet one more thing to add to the workload: A prompt declaration of emergency.