Forgive me;
IFR (below the "other" column on the Jepp SID) you have to make your chosen SID climb gradient, single engine, correct? Legally, maybe not, but if you wanna live to be an old pilot--yes. VFR, it's see and avoid, right y'all? There are many stories out there that we all have--and that's how we learn, by others' experience. So by all means, talk away, it's more fun that way--here goes my attempt:
One time, at band camp, I was flying outta OOBI (gulf), P-3 at night, very heavy, and it was hot and humid (of course). The aft observer reported sparks coming out of the engine, and we had to shut it down (it was safe to do so--we had altitude--which was nice). We dumped fuel, came back, and made an overweight landing (which was ok to do in an emergency 3 engine landing). If not shut down, the engine would have ate itself very soon, the mechs told us the next morning via the borescope inspection. Scary stuff.
I also had an engine roll back in a kingair to min fuel flow just before takeoff when I was ligning up on the runway in KBOI. If that had happened just a few seconds later, it would have been interesting. We aborted and had to taxi back pretty much single engine.
These two situations, in an IFR environment would have resulted in complying with the SID climb gradient single engine. Don't bet it won't happen to you. I have more examples, that time I hit 7 ducks at once in a P-3 while practicing a engine failure after V1 comes to mind. The airplane shuddered. All the maintainers on the ramp stared as we came to a stop in the chocks, guts everywhere. Damn birds.. that was unexpected. Aint that the way it is... Passengers will wait when it's a safety/legal issue. Pick your chief pilots brain, ask the other guys what they have done, and don't be shy about it. You know how pilots are, we love to talk airplanes--all of us. No worries.
The dark, ugly part of this business is when you have to make a safety decision (backed up by regs/ actual safety issues, aircraft limits, your limits, etc...) and your company doesn't back you up. None of us want to delay or cancel a flight, but when you have to, so be it. PAX get home itis or the commpany telling you to go when it's not safe to do so happens quite alot, and you have to stick to your guns. I didn't like that part of the business.
Fly Safe,
SR