However the government explicitly stated that Philip wasn’t a prince and it took a lot of back-and-forth between Elizabeth and the government to give him a princely title.
Here are the correspondences surrounding the upgrading of Philip’s title.
I didn’t see the government insisting Philip wasn’t a prince or resisting upgrading his title in the correspondence.
The file shows the Lord Chancellor, Viscount Kilmuir, writing in 1954 that Philip might already be legally a prince (though Kilmuir believed he probably was not). Kilmuir pointed out inter alia that Philip was styled “Prince Philip” in the 1948 letters patent on his children’s titles.
”4. A further question that must be considered is whether the Duke is now a Prince or not., and this is a matter on which there is some doubt. […] Some said that he was not a Prince because when he became naturalised he renounced his Greek and Danish Royal styles: others said that in spite of naturalisation he remained a Prince. The confusion was increased by the fact that in various formal documents he has been differently described:
1. In the Letters Patent of 1947 conferring the style of H.R.H. upon him - "Sir Philip Mountbatten, Knight of Our Most Noble Order of the Garter, Lieutenant in Our Navy".
2. In the Letters Patent of the 22nd October, 1948, conferring style and title on the children of the marriage of Princess Elizabeth and the Duke - "His Royal Highness. Prince Philip”.
3. In the Regency Act, 1953, and the Letters Patent of the 20th November, 1953, appointing Counsellors of State during the absence of the Queen and the Duke on their Australian tour — "His Royal Highness Philip, Duke of Edinburgh". This is the description that the Duke himself used when registering the birth of Princess Anne.
[...]
The objections were to specific formulations, not to the general idea of making Philip a Prince.
“Prince of the Commonwealth”, suggested by Queen Elizabeth II, was rejected by some of the Commonwealth governments.
“Prince Consort”, suggested by then Prime Minister Winston Churchill, was rejected because Philip did not like it.
Other proposals mentioned in the correspondence:
Philip proposed “Prince of the Realm”.
Prime Minister Winston Churchill proposed “The Prince”.
Queen Elizabeth II proposed “Prince of the United Kingdom” and “Prince Royal”.
Prime Minister Harold Macmillan proposed “Prince of the United Kingdom of Great Britain and Northern Ireland and Her other Realms and Territories”. This proposal later dropped the words “and Her other Realms and Territories” to avoid consultation with other Commonwealth governments.
[...] the substantive peerage of The Duke of Edinburgh superseded any Princely title. In fact, that is the truth of the matter and has been the case for a few hundred years now. Prince/Princess is something you use when the person in question does not have any substantive peerage of their own.
It is a bit bemusing that so much paperwork and time was dedicated to legalizing Philip's then-unofficial title of Prince (which was already used by the general public to refer to him, as the correspondence indicates), only for Buckingham Palace to continue referring to him as simply "HRH The Duke of Edinburgh" for the rest of his life. Only after his death have they begun referring to him as Prince Philip, Duke of Edinburgh.