Regarding the island departure; of course the straight ahead procedure could be used, but then this would have to be stated on the plate / in the company's SOP. What I am trying to say is that it isn't up to the pilot right there and then to decide what to do in the event of an engine failure, it must be carefully calculated beforehand in the form of an approved / published procedure.
Where I fly, the pilots don't have to think about it bacause the Flight Support department have issued engine failure procedures (plates) for every AD we use. We just study these before taking off...