[Michele]

But why exactly does the corporation need to officially cliam sovereignty?
If as a pragmatic business entity it can be content with de facto sovereignty, then the way to go is to follow the recipe provided by Forsyth in his The Deceiver.

Find a small overseas dependency of say France or Britain, a small non-strategic place. Those islets presently are satisfied with their status. But the corporation can pump money into an "independence movement". Make sure it gains traction with the small and unsophisticated local population, by showing that there's much to gain economically. Force the colonial power to hold a referendum and then free elections. Make sure the party that wins the elections is a tool of the corporation.

This is going to be yet another stamp-sized statelet, and formally a democratic republic, nothing new or objectionable. OTOH if it was the first corporate state in history, it would be quite objectionable. The former colonial power will be the first to recognize sovereignty of the new state and acknowledge the new democratically elected government, and other powers will follow suit.

Pay the politicians and do improve the economic situation of the general population, so that everyone's happy. You're done. You can have the corporation set up its HQ in the new state, CEOs are issued diplomatic passports, and you can even equip the state with a small but efficient brand new navy, if that makes you feel safer. Don't overdo this; nobody will object if the statelet is more than able to defend itself, but people in intel agencies would begin scratching their heads if it tried to purchase a nuclear-powered carrier with its air wing and escort warships.