How would we treat Klingon?
It is in the language list for works, and one Klingon artist is on site.
Klingon Pop Warrior
I don't know if there are any other invented languages available to us.
So far, there are (only) eleven Klingon(ian) works, see Klingon, which match the ones by Klingon Pop Warrior you added. Possibly, there are a few more around having been attributed to other language categories.
There are indeed songs in other artificial languages, f. ex. Kobaian (invented by Magma leader Christian Vander but not in the drop down menu), or Esperanto (in the list but so far not selected).