Gollum and Valinor...

[Bruce MacCulloch]

Though I don't know if the fate of Hobbits after death is ever addressed anywhere by Tolkien.

It is, in a roundabout sort of way. Professor Tolkien makes it fairly clear in several places that Hobbits are Men. Therefore, they would most definitely be under the same Doom as Men.
In a footnote to Letters no. 151, he says

The Hobbits are, of course, really meant to be a branch of the specifically human race (not Elves or Dwarves) - hence the two kinds can dwell together (as at Bree), and are called just the Big Folk and Little Folk.

Also in no. 319:

...and that my 'hobbits' were in any case of wholly dissimilar sort, a diminutive branch of the human race.

In Morgoth's Ring, chapter X, "Of Dwarves and Men":

Hobbits on the other hand were in nearly all respects normal Men, but of very short stature. They were called 'halflings'; but this refers to the normal height of men of Numenorean descent and of the Eldar (especially those of Noldorin descent), which appears to have been about seven of our feet. Their height at the periods concerned was usually more than three feet for men, though very few ever exceeded three foot six; women seldom exceeded three feet.They were not as numerous or variable as ordinary Men, but evidently more numerous and adaptable to different modes of life and habitat than the Drûgs, and when they are first encountered in the histories already showed divergences in colouring, stature, and build, and in their ways of life and preferences for different types of country to dwell in (see the Prologue to The Lord of the Rings, p. 12).