There’s no particular hand wringing over the name in the UK. Charles I may have been a disaster but Charles II was a pretty solid monarch (despite his lothario ways).
Every ‘kingly’ name we’ve had has some dodgy monarch attached to it somewhere down the line (Richard, Edward, Henry, George, William, you name it).
And, as noted, if he took Charles, it would mean that “Charles III” means him, and delegitimize the use of “Charles III” to refer to the “Young Pretender”. Not that after over 200 years we should give a crap about Jacobite claims, but it has entertainment value.
“Philip” would actually be entertaining, too. We would get to watch the hand wringing over him taking a “foreign” king name, which was all well enough for France and Spain, but isn’t a name for a BRITISH king.