Please just use Kerberos instead of fiddling with uids. It’s the only sane way to get NFS access controls and user mapping. Works on both Linux and macOS (but there’s no NFS on Windows anyway).
I’d say you can run the Kerberos KDC on the NAS but if Synology has some locked down special OS you’ll need another machine for that (edit: but you say you have other servers already so that shouldn’t be a problem).
Unfortunately SMB is so screwed that you can’t reuse ordinary Kerberos for authentication there, which is unfortunate if you want to have both that and NFS. I’ve yet to look into whether Samba AD can be used for both.
True. I knew I should have left that as “NFS 4” because someone would comment this. From what I’ve read (never used it), NFS 3 is very different to 4 and also just kind of not worth using, especially just for Windows, since it has no security at all.