I think that Charles will become Charles the Third, the most important reason for me is that back in 1948 Prince Philip and Princess Elizabeth have chose that name for their firstborn and heir. That was a deliberate choice and undoubtedly the parents will have been aware about the two Kings with that name in the history of the British monarchy.
Last year the Netherlands welcomed a new King. In the television interview before the Investiture the then Prince of Orange made clear he would be King Willem-Alexander
. Not King Willem IV
or King Willem IV Alexander
. He stressed that Willem-Alexander is his given name, that is how people know him all his life and he saw no reason to change it at all.
It is very well possible that the Prince of Wales has exact the same idea as his (then) Dutch colleague on this topic. Charles the Third
, it sounds great. It links to Don Carlos III de Borbón y Farnese, King of Spain, King of Naples and Sicily, Duke of Parma and Piacenza (1716-1788) widely seen as one of the best Kings of Spain ever, promoting science and university research, facilitating trade and commerce, modernizing agriculture and avoiding wars. May Carlos III be a great example for Charles III !