I think the "bath" is the long "a" vs short "a" split[0] (usually associated with north of England vs south of England) rather than r insertion. There is however something called the "intrusive r" among "non rhotic" speakers "such as those in most of England and Wales" [1] which I learned about thanks to this recent research, where "r"s are inserted where they don't exist, e.g. "supernova-r-in the sky" and "law-r-and order".

[0] https://en.wikipedia.org/wiki/Trap-bath_split

[1] https://en.wikipedia.org/wiki/Linking_and_intrusive_R