By all means you can mention it, but it is their prerogative whether they take it on board.
Teachers cannot please all parents. They will split classes based on various factors:- friendships (good and bad, some children will be kept apart for their own good, despite being 'friends'), abilities, gender, EAL, pupil premium etc. The 3 classes will need to be as equal as possible in each of these categories. Friendships are definitely considered but can't necessarily be the be all and end all over other factors.
At the end of the day, the children will still see each other on the playground and they will make new friends within their own class. As long as they move up with some friends, it may not be the current 'best' friend and that's ok. At the end of the day, the teachers actually know who plays with who in school better than the parents. You only know what they tell you. They're seeing it every day.
Look on this new friendship as a positive that your previously shy boy is branching out and capable of making new friends. He will manage to do that again, if moved away from this new friend. To be honest, you're only taking his word for it that he is now very close to this new friend and doesn't play with the old one at all. If it's a strong friendship, school will definitely be aware. They may have only played together a few times though. Young children's definitions of 'best friend' can be very different to our's. If it's a very new friendship, at that age it can sometimes be a case of out of sight out of mind. He might not necessarily be bothered if he's not with the new friend. If he no longer sees them in class, he'll move on to someone else. Young kids can be fickle like that. The old friend may become flavour of the month again if they're still together, or he may move on to someone else entirely.
As I said, by all means you can mention it Be mindful though that school will probably be having similar requests from 20+ others parents in the year group. They can listen and consider requests, but can't accommodate all requests. The decisions they do make will ultimately be made in the children's best interests, based on their knowledge of them.
To be honest, their new class lists have probably already been decided on anyway, weeks ago.