I'm thinking of something like "alien" or "gentile", that doesn't sound/read as clunky as "non-elf".
There is none.
The relevant text is the essay Quendi and Eldar, published in History of Middle-earth 11: The War of the Jewels.
This is a lengthy essay, with appendices, covering:
Origin and Meanings of the Elvish words referring to Elves and their varieties. With Appendices on their names for other Incarnates.
It also provides a good deal of history as interesting asides.
Section B, "Meanings and use of the various terms applied to the Elves and their varieties in Quenya, Telerin, and Sindarin", contains the following two statements:
Moerbin was similarly an equivalent for Avari; but that it did not mean only 'Dark-elves' is seen by its ready application to other Incarnates when they later became known. By the Sindar anyone dwelling outside Beleriand, or entering their realm from outside, was called a Morben.
The Men of the Three Houses were also soon removed from the class of Moerbin. (Note 8, p. 408) They were given their own name, Edain, and were seldom actually called Celbin, but they were recognized as belonging to this class, which became practically equivalent to 'peoples in alliance in the War against Morgoth'.
These are the only two terms given, which are explictly stated to be Sindarin, and as we can see they include Elves as well as other Incarnates.
Other notes in this essay cover Quenya and Telerin, noting that whereas 'quen' was a general word meaning 'person' and was used in compounds for other Incarnates too (examples of 'roquen, horsemen', 'kiryaquen, sailor' and 'arquen, noble' are given), 'Quende, Quendi' was exclusively used for Elves.
In contrast to Sindarin:
Moriquendi was now applied to all other Elves, except the Noldor and Sindar, that is to Avari or to any kind of Elves that at the time of the coming of the Noldor had not long dwelt in Beleriand and were not subjects of Elwe. It was never applied, however, to any but Elvish peoples.
The essay also contains three appendices which are relevant to this question:
- Appendix A. Elvish names for Men.
- Appendix B. Elvish names for the Dwarves.
- Appendix C. Elvish names for the Orcs.
While these appendices cover much additional interesting material, providing confirmation of a general term for all Men ("Firyar 'Mortals', or Firimar"), giving the Quenya "Attalyar", Sindarin "Tad-dail" term for "Bipeds" (from context clearly a term for unintelligent two-legged beasts), no overall term for non-Elves is given.
Although there are a variety of euphemisms applied to individual races by the Elves, I've never come across a word for "all non-elvish people" and, as far as I can tell, Tolkien never used one. We can make one up though, in one of the various Elvish languages. I chose Quendi, because it's the one I know best.
This uses the negative prefix ú- ("not-, un-, in-") and quendi ("speakers", the name Elves use to refer to themselves). It literally means "not-Elf" or "un-Elf". I'm not sure whether the Quendi would use such a word or come up with something more poetic (like they usually do).
As an aside, the ú- prefix has a complicated history. Tolkien wasn't sure if it should represent simple negation (as above) or something bad, evil or immoral, such as for the Úmaiar, the Maiar who followed Melkor, like the Balrogs. He seems to have settled on ú- being a negative particle in the end.
Correction on Úmeldar
I proposed Úmeldar in my comment above, but I'm not sure of its correctness anymore. Since a discussion followed the comment, I won't remove it, but I'll give more information here.
Eldar ("people of the stars") is mostly used to refer to the Elves who accepted to leave Cuiviénen with Oromë (contrast with Avari, those who didn't). Quendi is the correct word to refer to all the elves.
I also think it should be Úeldar, as úm- seems to be a verb prefix. I mentioned Úmanyar as another example of the úm- prefix, but it is in fact a contraction of Úamanyar. This word is composed of four parts:
- ú-, a negative prefix, like not-, un- or in- in English;
- aman, "blessed, free from evil", but also used as the name of "the Blessed Realm", where the Valar live;
- -ya, an adjectival ending (a way to turn a noun into an adjective); and
- r, a plural ending.
So Úeldar would literally be "those who are not people of the stars", which would not be an answer to the original question. It's also a made-up word, by the way: Avari ("recusant, one who refuses to act as advised or commanded") is used by the Eldar instead.
The Quenya word fírima means "mortal" or "those able to die." According to the Tolkien Gateway:
Mortals were any of the races of Arda who were subject to old age and death (as for example Hobbits), but applied especially to the race of Men.
The plural (the way to designate an entire race or collection of races) would therefore be fírimar.
This word is used mostly for Men, though it could be for other mortals.