I learned that e.g. while 「彼は食べていた。」means "He was eating.",「彼はもう食べていた」means "He had already eaten." In that case, does using もう in this context always transform the meaning from a continuous state to a resultant state, or is it ambiguous? That is, what if I want to say "He was already eating?" I guess I can say something like 「彼はもう食べ始まっていた。」 to be unambiguous, but is simply 「彼はもう食べていた。」 still correct?
Yes, it's ambiguous. To begin with, 食べていた can mean "he had eaten it" too, beside "he was eating".
(Incidentally, you don't say 食べ始まっていた but 食べ始めていた. 始まる doesn't have the auxiliary usage unlike 始める.)
Rephrasing もう食べ始めていた as もう食べていた is still correct.