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It's generally accepted that one of the best chances for a successful invasion of Great Britain is the French and Spanish Armada of 1779. Suppose there was an invasion that succeeded, or was at least good enough to rip Ireland away (through force or peace talks) as a separate country? France and Spain were both ruled by monarchs- can't see them agreeing to a republic. But who would be the new king?
Is there a local with a strong enough claim (at least backed by foreign guns)? Charles III (Bonnie Prince Charlie)? Or would they pick one of the underemployed princes available in Europe? I assume he'd have to be Catholic to be acceptable (and male).