I'd be inclined to stretch him sideways by doing non-school activities, rather than giving him additional curriculum-based work, tbh.
Music can be good for kids like this can he learn an instrument? Learn a bit of music theory? Or another activity that will satisfy the geeky side of him without making him even further ahead at school drama? photography? Dd1 at this age got a build-your-own camera kit and was obsessed for ages. There's a boy in my ds's class (v. mixed state primary) who is writing a novel, and intermittently a group of them try to write and film screenplays based on his 'book'.
Chess can also be good for the mathematically-inclined. Etc.
I've been through this and got the t-shirt with dd1, who was in quite a rough primary school which was not at all sympathetic to her needs. For various reasons we didn't move her, and actually I think having to muddle through and focus on her own interests rather than just swimming with the stream as she would have done in the more academic private schools we looked at has made her a more interesting, thoughtful and confident teenager than I think she might have been had she been in a more obviously 'suitable' environment. And she sailed into grammar school without any particular extra preparation.
DD2 looks set to be the same (in Reception, just starting to read chapter books, class still doing basic phonics sounds). It's not a problem because her teacher is fantastic at doing creative and interesting stuff with them, so she is in no way bored. But in a school that had a much more grindy, 3Rs approach, she might be bored senseless.
I think you need to talk to the school. If they get where you're coming from, are basically supportive and are providing an interesting environment for him and other bright children in the class, and (most importantly) he's happy, then I'd save your money tbh. If you get the feeling that the schoolwork is a bit unimaginative (too many dull photocopied sheets coming home), not sufficiently differentiated and the school do not react well when you try tactfully to discuss these issues, then it might be time to look more seriously at other options.
But don't panic or feel you somehow have to move him. There are loads of bright kids in state schools, having a fun and interesting time, doing well academically and getting into good secondary schools and universities. They won't have the same glitzy facilities, they won't be pushed quite as hard and they won't have as much homework (this is a plus, btw), but if they're happy and well-supported they'll be fine.