Do you have to date Jasper to get him to stay on the island?
No, you're supposed to be able to make him stay without dating him. If you can't, well, maybe you found a bug! Let me know :)
Every time I talk to him I just get the date request. I waited until pretty late to finish the quest with the original fruit stand girl if that's what caused this.
Hm, sounds like a loophole! I will take a look at it tomorrow :)