[Mithalwen]

It isn't quite presented as fact in the Appendices.I don't know if further mention is made in HoME to clarify.

In the tale of years in LOTR it says 1541SR Legolas built a grey ship in Ithilien, and sailed...over Sea. And with him it is said, went Gimli the Dwarf". However there is a note at the end of "of Durin's folk which purports to be a late note in the Red Book.
"We have hear tell that Legolas took Gimli ....if this is true then it is strange indeed that a Dwarf should be willing to leave Middle Earth ..or that the Lords of the West should permit it. But it is said that Gimli went also ...to see ..Galadriel and it may be that she, being mighty among the Eldar, obtained this grace for him. More cannot be said of this matter".

So the "translator conceit" allows Tolkien not to give a definitive answer. The note on sources in the Prologue increases vagueness. The Red book was copied annotated and what became the appendices gathered from various sources long after the event (the time of Peregrin's great grandson). I think Gimli would have been 262 when Aragorn died which is a good age for a dwarf - he may have returned to Erebor to die or it may be that the grace was granted him because he had so little time left.

Other than Tuor who is a special case and Amandil whose fate isn't known, Frodo is the only mortal with much natural life span remaining who was admitted to the Undying lands and he was a special case. He had pretty much died after the Morgul attack in the sense of living fully in the world of Middle Earth. Remember Gandalf's thought about him becoming like a glass full of clear light and noticing a transparency about him in Riendell? Which itself was soon after explaining why Glorfindel became a shining figure as he faced the NAzgul - that he has dwellt in the Blessed Realm and lives at once on both sides. Tolkien's decision/realisation that Glorfindel of Gondolin and Glorfindel of Rivendell were the same means that Rivendell Glorfindel had died and been sent back to aid the struggle against Sauron. Another exceptional situation and the juxtaposition maybe not a coincidence.