Actually, "I became a big fan of his" is natural, and it is known as a double possessive. It has been idiomatic in English for decades. Keep in mind that it is only used when the second part is a person, you wouldn't say "I'm a frequent visitor if the museum's". However, you should avoid double possessives in official documents.

A)I became a big fan of his ______. You need to put something here, like "I became a big fan of his work". B)I became a big fan of him. This is correct. C)Who is your favorite movie actor? For me, it is Daniel Day Lewis from the UK. D)Who is your favorite movie actor? For me, it is the UK actor Daniel Day Lewis. <- I think this one sounds better, but I think C and D are both OK.

For me, it is the British actor Daniel Day Lewis Just like you say "American actor" and not USA actor