I know this situation is far from perfect and there's no obvious solution.
But I strongly believe that trading Tavares will result in losing him for good. He doesn't have a good reason to stay. The team is a mess more often than not, coaching situation is... well, let's just say I would love to have a serious talk with Weight, the GM is completely incapable of playing with the big boys, there's also the arena thingy and probably something that I'm not remembering right now. So right now the smart thing for Islanders would be to hold on to any kind of advantage they have. And that would be the fact they can offer him an extra year and the fact that they could be loyal to him the way he is loyal to them (he is, really, and if someone doesn't see that they're beyond help).
Trade him and you lose that.
The thing is, Tavares does not have a team that could take the weight of things when he's down, an advantage that many great players on other teams have. So when he is a bit down, when he plays a bit worse than in the begining of this season, there's no one to step up. I mean, Barzal kind of was, but I think Weight broke the boy, hopefully temporarily.
And I don't know if this is some kind of defence mechanism, but I really can't see the love the Islanders' fans have for Tavares lately. I know, I know. Crappy situation. But even if he's not traded, even if he signs with other team as an UFA, good lord, the guy gave the organisation and fans 9 years of his life. Was really all of it so bad that they can't spare a bit of love and respect for him?