I think he might have been trusted, and indeed, that he would have been rehabilited. If he managed to get past his old self, and mainly, with the Ring finally being gone - that would be a relief. And I find the idea of Sam Gamgee living in Bag End with his family and Sméagol working on the lawn outside the house quite cute
(in Bag End! Not Rivendell - see below why.)
However, there are a few things which make me doubt. First, sailing to the West - indeed, I am really not sure if he would have been admitted, even to some "healing stay" on Eresseä. Let us not forget that he was a murderer. That does not quite fit, and I get the feeling that he won't get away with it so easily in Arda. Another thing is if he himself would want to be there: I mean, staying among the Elves - not sure if he will like it. In the Shire, perhaps, yes, that's possibly the only place where I can imagine him living. But among the Elves?
One "smooth" alternative would be that at the same year when Frodo will be leaving Middle-Earth, Gollum would simply die of old age. Why not: he was old enough, far older than Bilbo, for example. He would die peacefully, among the hobbits. That will be also a part of the signal for Frodo to leave these shores at last.
One side thing - I find it interesting to think of this; you have to consider that Gollum would have probably met Bilbo again, had he remained alive. That might have been nice, don't you think? A bit weird, perhaps, but...? I imagine it in a similar way as I imagine Frodo might have returned to good terms with the Sackville-Bagginses after his return home.