Sorry to disagree, cemetery friends, but I have library cards for 3 local authorities but I am only resident in 1. I find that all 3 libraries have different selections of books.
The nearest library to me, about 2 miles, is in a different local authority.
My own local authority library is about 3 miles away.
I have had no problems signing up to the library services with any of them. I also have no problem popping into any of them and using their internet services.
My daughter joined a neighbouring authority library this summer because they have a sheet music department, again no problem.
I suggested yesterday that you give them a ring. My advice still stands.