In 1960, Queen Elizabeth II declared in Council that she and her children would continue to be known as the "House of Windsor", but that her direct descendants, other than those who are HRHs and married female descendants and their respective descendants would have the name Mountbatten-Windsor. The full text of Her Majesty's declaration is reproduced below.
In practice, it seems that even the Queen's children, when they need to use a surname for any reason, use the family name Mountbatten-Windsor, which seems to be inconsistent with the Queen's declaration that they should continue to be known as the "House and Family of Windsor".
If anything, I believe Charles could rectify that inconsistency making it officially known that the Royal House should be known as the House of Mountbatten-Winsor.
It was the queen's wish that the family continues on as "Windsor". But while this is binding as long as she lives, it need not be binding for king Charles. He studied history, he is proud of his father's family (we've seen that on many Greece-visits), he taught his sons to be proud of that, either.
So he must be aware that "Mountbatten" is just the English translation for a name (Battenberg) given to the children from a unequal marriage between a Hessian prince (whose actual birth might have been not legitimate, as there were strong rumours he was conceived on the wrong side of the blanket and only legitimized at birth) and a lowly noble lady. Through marrying back into the Hesse-family (Prince Louis Alexander who married Victoria of Hesse, daughter of Queen Victoria's daughter Alice)) and into the (BRF) Windsor-family (his brother Henry who married queen Victoria's youngest daughter Beatrice) both Battenberg-brothers became British subjects and were considered part of the BRF, so in 1917, when the Royal House changed name to Windsor, their name changed to Mountbatten and they became Marquess of Milford-Haven (Louis) and Carisbrooke (Henry).
So neither the Battenberg nor the Mountbatten-name is a legitimate Royal family name nor is it the name of Prince Philip's paternal family.
Problem is that Philips paternal family never had a "family" name. They only ever identified via their basic territorial rights as was usus in Germany. The dynasty originates from Oldenburg in North Germany. The first known count of Oldenburg lived around the time of William the Conqueror. Similar to that Prince Albert's "Saxe-Coburg-Gotha" is not a name, but a branch of the dynasty of the Lords of Wettin (another territorial name). So in fact, neither Charles' father nor mother have a "family"-name apart from Windsor which was "invented" for exactly this purpose in 1917.
So I wouldn't put it past Charles to let go of the Mountbatten and to stick to Windsor in case a surname is needed.