It is an acronym and typically those are all caps but in this case “bae” is used as a term of endearment to address someone or refer to someone so “My bae loves that song” or “How are you doing, bae?” are accurate. An acronym for a place or a subject like LSU (which is a college) would be referred to in all caps “I graduated from LSU”

i always thought 'bae' was short for 'babe'. in that case it is, indeed, "bae".

Maybe I'm old, but . . . . what's a BAE?

I don't think it is a word. I would assume it to be a regional or ethnic variant of "baby".

just like "lol"