Very nice one! And good to see this thread back up...
I would say the answer is Faramir.
Brother killed with Androg's fate; - Boromir, by arrows, like Andróg was
Father broken with despair and regret. - Denethor, of course
I had to choose between
Approval and my heart that was keen. - the question of Frodo and the Ring
A descendant of mine was named
After one by the miserable betrayed. - if I recall correctly, Faramir's son was named Barahir, who was betrayed by Gorlim
The one who was named after me
Has a relative that was dived for in the sea. - I am sort of wondering about this one. Obviously the person in question is Faramir, son of Peregrin; but whoever is his relative remains a mystery to me. There are those Tooks spoken of as going to the sea, but that does not really sound like the case, also the relationship is rather distant. Unless, I think there is some Took whose name was Pearl, so if it is metaphorical... I don't recall how distant such relationship would be, though, either.