"the fact that in TT and ROTK Tolkien splits the action into Book 1 and Book 2, with Book 1 recounting the war and Book 2 recounting the quest, suggests that they are equally important parts of the overall story"

Umm... no... Tolkien wrote the story that way simply because he knew of no other way to bounce back and forth between the two... cause really, he was never a very good writer...

And as for the 'splits' JRR had little to do with them... That was Unwin who spilt the book Lord Of The Rings, into the current 3 books, with 2 'sub-books' each...

"Tolkien's experience of war was one of the major ingredients he used in his fiction"
Not consciously... Tolkien HATED allegory...