I don't think he would want to abdicate but it wouldn't surprise me if there's some exaggerated scandal (maybe something with the Church of England, since Charles isn't too crazy about being Defender of THE Faith, or some meddling in politics) that makes it untenable for him to stay on the throne.
If the monarchy and the church came to blows in modern Britain, the church would almost certainly lose the battle. Even against Charles.
Also the idea of him going with a different name kind of funny. He's been know as Charles for 73 years, they'll still refer to him as "King Charles" even if he goes as George VII
Yeah, and people overstate the degree to which it's normal for monarchs to change their names. The only one who did anything really weird was George VI, who was never publicly associated with the name George until the first day of his reign. Victoria's first given name was Alexandrina, but she was already known as Princess Victoria of Kent before her reign, and Edward VII was always double-barrelled as Albert Edward to differentiate him from the late Prince Consort and his eldest son (Prince Albert Victor).
Well, George VI was known as Albert (or Bertie) before becoming King. But Albert is a sacred name (for some reason) so King Albert was out of the question. In honor of his father it became George.
Charles will be King Charles III. It would be ridiculous to do anything else.