I’m unaware of any restriction on the name Arthur. It has show up a couple times in British royal names.

Henry VIII named his oldest son Arthur. So we’ve had an Arthur, Prince of Wales already. Unfortunately for Arthur he died when he was 15, from some unknown disease. Henry VIII inherited the crown instead of his brother.

Queen Victoria also had a son she named Arthur, or more precisely Arthur William Patrick Albert or Prince Arthur, Duke of Connaught and Strathearn. As the third son, he was unlikely to inherit the crown, but no one seemed to have objected to the name.

So historically there hasn’t been a ban on the name. It just hasn’t been used very often. Of course current debates over names are bit out of scope for this sub. But even if they were to name the kid Arthur, and he made it to king, that wouldn’t mean he would reign as King Arthur. Kings have at times taken a different name than their first name. Edward VII for example was born Albert Edward. So a potential King Arthur might instead choose one of his other names as his reginal name.