The title 'Return of the King' screams 'the return of Jesus Christ'.

I don't think it does. I think it screams just what it says, the return of the king of Gondor to the throne. After all, the return of Jesus Christ is meant to represent the end of the world, and the return of Aragorn as King of Gondor and Arnor may trigger a new age, but not the end of the world (except as it is relevant to the older peoples like dwarves and elves). One may as well say that "Welcome back, Kotter" is an allegory for the second coming.

Gandalf's resurrection echoes Jesus' own.
No, it doesn't. Gandalf didn't die to save the world from sin. If anything it represents more, in anthropological terms, a passage of rights, where he endures a trial and emerges purer and stronger. In fact, Gandalf is specifically banned from saving the owrld. The most he can do is assist those who seek to save it.
The Ring can represent sin.
Surely temptation, not sin? Sin is done by a person, knowingly. Once tempted, one may sin, but the temptation is not a sin. Sin is a free choice to deny god, or to act contrary to god's will.
Sauron can represent Satan.
I thought that was Melkor's job? Sauron may be a manifestation of a devil, and agent of evil rather than The Devil.
However, the great thing about Lord of the Rings is that it's not strictly allegorical. You may see one parallel, and another, but together they say something else. It's great. There is so much knowledge to be gleaned in between the lines.
I consider The Lord of the Rings to be a 'Christian' work, because they've greatly helped my in my walk with the Lord.
So Buddhist texts that enable you to do the same thing would be "Christian" works? Why not call it a book that has meaning in your life? I may as well call it an atheist work because it helps me not be a Christian, but I don't presume so.