This issue arose in the 19th Century with regard to James Brooke, the first Brooke Rajah of Sarawak (r 1841-1868). On the advice of the Foreign Office, successive British Governments refused to recognise Sarawak’s independence under Rajah Brooke’s rule on the grounds that British subjects could neither repudiate their allegiance to the Crown nor become sovereign rulers themselves.

This advice was incorrect. Under the terms of the Sophia Naturalisation Act 1705, the Electress Sophia and her Protestant descendants were deemed to be British subjects. By the time of the Act’s being repealed by the British Nationality Act 1948 (most, if not) all of Europe’s Protestant monarchs were, under the terms of the Sophia Naturalisation Act, British subjects. Any Protestant descendants of the Electress born before the repeal of the Sophia Naturalisation Act in 1948 still are. (The British finally recognised Sarawak’s independence under the rule of Rajah Brooke in 1863: the USA had recognised Sarawak’s independence in 1850.)

Since the British allow for dual citizenship, your question turns on whether the Vatican does. That, I don’t know but, given the cosmopolitan nature of the Vatican’s Government, I would be surprised if it did not.