I believe your first option is the best but with a little modification. Moses originally had an Egyptian name that sounded almost exactly like a Hebrew name.
The pun involved in the name is elaborate and crosses languages. Names like Tutmose, Ramose, Amenmose are well attested from Egypt. The addition of -mose makes it "born of Amen," "child of Tut," or "shaped by Ra." Obviously, these are names of piety, showing the child's attachment to the patron god. The final "s" on Moses (Hebrew Moshe) comes from Greek which does not allow masculine names to end with a vowel (Yeshua becomes Jesus, Moshe becomes Moses. You will also note that the Hebrew "sh" in both became the Greek "s" as koine Greek does not have the "sh" sound).
An oddity is that the Hebrew name is active (Moshe) instead of passive (Mashuy). One might expect that his being drawn out would result in his name meaning "one who is drawn out" instead of "one who draws out." However, this is a wordplay, not a precise description. The exegete should not try to be more precise than the original author or speaker intends to be.
It would be unlikely (but not impossible) that the princess would know the language of the slaves. However, everything she has already said has been recorded in Hebrew. Most likely, her words were in Egyptian and translated into Hebrew (some might argue that she said little beyond the name and the rest of the phrase was placed in her mouth by later writers, but this seems unlikely).
Having found the child in the Nile, the source of life for Egypt, the princess could easily have seen the child as given by divine providence. It is possible that she gave him a longer name (perhaps Ramose after the sun god or Hapimose after the god of the Nile).
Translating the princess' words allowed for a sophisticated pun. The name brought to mind his later works and the translation was rather free (as it often was in those days). Perhaps she said in Egyptian, "I will call his name 'Mose', for he was born from the water." The the Hebrew pun is natural and requires only a small modification to her words when translating: "I will call his name 'Moses,' for I drew him out of the water."
Where the Egyptian means "born," it sounds very like the Hebrew "drawn out." "The one who draws out" is how Moses has always been remembered (Isa 63:11). The princess drew him from the river to give him physical life. He drew the Israelites out of Egypt through the water and gave them spiritual life.
He was born with a great destiny, and even his name and circumstances of his naming showed that.