Is there a more specific word for endonyms which simply mean “our language” or similar and are semantically awkward for outsiders to use?

Not aware of such a term, but it’s a nice distinction: the endonym is really just a pronominal reference, so much “ours” that it doesn’t warrant a name at all.

I could coin the term hemeteronym, “ours-name”, for it, but I won’t. It’s a pronominal, or deictic, endonym.