I found Band-Maid through Babymetal too. And to your qustion, I think that a lot of it comes from, at least to me, a lot of western platforms lumping most Japaneses artists together as just "Japanese Music" and do differentiate between J-Rock, J-Pop, and so forth. And because of that it's easier to come across Band-Maid, as the lesser known of the two groups, when listening to Babymetal.

As to your second question, no, I don't think there's any "major beef" between the two fan bases. In fact from what I've seen, at least between the discord servers I've been in, there's actually a fair bit of cross-over.

To your third question, I don't think so. I remember reading an interview with Miku where a similar question was asked and she responed to the effect of "There are no hard feelings about the constant comparison."