What bothers Joy about Meghan isn't her politics. It's her lack of any sort of sense of humor. Joy gets along fabulously with Ann Coulter because Ann has a clear sense of humor.
Meghan has this worldview where she's the life of the party and wants SO badly to be loved by Joy because that would validate her perception of herself - that she's loveable and fun. I think Meghan's whole life is about seeking that approval. That's she's fun, loveable, a vital political analyst, etc.
As a comedian, people like Meghan are the most fun to take down because they can't laugh about themselves. Joy really has to hold back because she knows that Meghan can't take it, and she does have to work with her.
Meghan is the true definition of a snowflake.