I'm thinking either like a polarizing rock vs disco situation in the 70s or how country music is much more popular in the south.
Also can't give a definitive answer, but here are a few things to consider:
The Beatles were hugely influenced by Elvis and were teenagers during Elvis' early peak years. They were all 20-22 at the time they hit massive success in America, meaning that Elvis' fans from his early days were probably all in that same age range.
Elvis, at the point of the Beatles breakthrough in the US in 1964, was focused on film and his success was slipping.
Most likely, the only divide would have been teenagers who loved the Beatles v young adults who had loved Elvis as teenagers. But, the Beatles popularity likely also affected the slightly older generation and they would have been fans of the band as well. I'm sure people could like both at the same time, especially since the Beatles thought so highly of Elvis and his music was of considerably lower quality at this point.
Can't give you a definitive answer, but maybe I can point you in the right direction:
The Question you're asking is something that cultural geographer would be more likely to look at than a historian, I think. One of my professors produced a map of propensity to purchase Elvis memorabilia by county. Unfortunately, I believe he did so befor the era of digital publishing. I got one look at a decade ago, so I can't give you an exact description, but I remember there was a very clear regional variance. If there is a sub like this, but devoted to social science, you might consider asking over there. Please excuse any typos, as I am on mobile.
Your question doesn't really make sense. Can you explain why you think there would be? Your question seems premised on the idea that Elvis and the Beatles exist in different categories and so would attract different fans (like rock and disco), but the Beatles themselves were enormous Elvis fans. All of the British Invasion bands were enormously influenced by rockabilly and blues.