http://tcrf.net/1943:_The_Battle_of_Midway_(NES)
This page needs a lot more research contributed to it, such as whether or not any unused music might appear within the game's ROM. I am fairly certain there just might be an NES version of the BGM we've heard on the arcade version that we first hear in rounds 3 and 4.
The NES version of the game was later released in Japan as 1943: The Battle of Valhalla.
All this page has for now is this morse-code soundbyte, which is described as playing twice during the game's intro sequence. Converted, this code reads "CQ DE CAPCOM DN SK" and translated, it reads "Calling any station. This is CAPCOM. Down and out."
~Ben