"Use your airmanship! What do you think you're being paid a small fortune for?"
More realistically: to decide if you have a fighting chance we would need more information on the SID and the engine-out SID. How many seconds, miles or minutes from airborne is the emergency turn? Is there manoeuvering room to turn back? The late engine failure should mean you are above the profile at that point. How much further have you got to go to MSA? And so on...
Private jet is right. For example, think of the all-engines-failure situation.