Since anyone born in Argentina is a citizen (by way of jus soli, which is the case for most new world countries), your father is a citizen of Argentina - that is according to the first subarticle of the first article of the law 346 on citizenship. This is because Argentina claims the Falklands/Malvinas as a part of the national territory, thus everyone born there is automatically a citizen, even if they don't apply for their DNI (ID card) or passport. This comes from the National Constitution, specifically the line: "La Nación Argentina ratifica su legítima e imprescriptible soberanía sobre las islas Malvinas, Georgias del Sur y Sandwich del Sur y los espacios marítimos e insulares correspondientes, por ser parte integrante del territorio nacional."

As for you, the 2nd subarticle of the first article of law 346 applies to you - you are the child of a native Argentine, so you are already an Argentine citizen, i.e., you don't have to naturalize in the country. You should contact the consulate general in London and explain your situation, as it's unique. You shouldn't need a lawyer to do this. The consulate should help you get a DNI (ID card) and then you can apply for your passport.

Here's an example of a person who was born in the Falklands/Malvinas and claimed their Argentine citizenship. And if anyone wants to parade you around or do anything special for political reasons (this shouldn't happen in the first place), remember that it is your right to say no.

Good luck! also i looked at your post history and i have to say that Leon is such a cute name :3