I think the somewhat confusing thing here is that Japanese can pronounce the same kanji in different ways, from 'purely local Japanese' to variants of old Chinese pronunciation.

米国 "beikoku" sounds like a strange choice for "America". But then in 亜米利加 it's pronounced "mei", which does indeed explain the choice better.

Chinese use 美 'mei' for America and so 美国 'meiguo' is the US.