I decided to try split DNS to avoid leaking the internal IPs, but it turned out a bit more fragile than I imagined.
Especially Android is finicky, ignoring your DNS server if it doesn't like your setup. For example, if it gets an IPv6 address, it requires the DNS server to also have an IPv6 address, or it'll use Google's DNS servers.
It works now but I'm not convinced it's worth it for me.