I don't know if that's right, but your suggestion is another story. But let me tell you something, the sentence above means, you're eating "with them" at breakfast, not "eating them". But I believe that "with them" doesn't only mean "with other people", for example, it can also mean "with foods/food items on the table other than the main breakfast itself". However, I still appreciate your suggestion.
That would mean "I eat breakfast with them", not "I breakfast with them", but it should be fine.
LOL gotta love the "A Game of Thrones" reference. I think I've heard of "to break fast" from other novels as well. Never heard it in modern usage though, nor have I heard of the verb "to breakfast," but I'm going to start using that one now.
I found a reference to "To break one's fast:" https://en.wiktionary.org/wiki/break_one%27s_fast
Actually it isn't trying to reference anything; the actual Esperanto usage is very common.
"Matenmanĝi" means "to eat for breakfast", not "to break the fast", though if you try to translate the word literally into English it might seem that way. You''ll find that Esperanto often expresses with a single verb or adverb something that English would express with a phrase.
What is wrong with "I have breakfast with them" ? It seems to me that "To eat breakfast" and "To have breakfast" is the same thing. English speaking people (at least Canadians) would rather say "to have breakfast". By the way you can "have breakfast" but not necessary "to eat it".
Because "kun" is an ordinary preposition and you'll never have an -n ending with an ordinary preposition.
More info on the two kinds of prepositions here: https://blogs.transparent.com/esperanto/keys-to-understanding-esperanto-prepositions/