Bit of a catch 22, living in paradise but no time off!
Many years a go we used to handle Air Jamaica and one of the stipulations was on the amount of rest we had had before working on their a/c, so clearly in the Carribean there are regulations, but which are you governed by, local or the airlines?
Could you not get somebody else to do the odd transit once a week, I mean somebody like VS or one of the other airlines, it would involve a contract and some payment of course.
At the end of the day what the airline is asking you to do is not legal from the EASA side of things within the directive as that is what they are goverened by.
Any way what happens when you go AOG, spend 16 hours trying to sort it then the next 5 hour transit turns up.
How do you maintain ETOPS when on your own, some things require independant inspections, a pilot can only do some things!
You need to talk to the airline.