Wireguard issues / confusion

I made a whole write-up on this forum describing one way to do this:

My post uses IPv6 ULA addresses but it works the same way with IPv4. You basically create a subnet plan and second-level domains so DNS forwarding works.

Your situation is slightly different however, it appears you want a site-to-point tunnel instead of site-to-site. In that case the client needs to support "conditional forwarding" or "split DNS". In these systems, the client runs a DNS resolver locally and forwards DNS queries through your Wireguard tunnel as needed on a domain-by-domain basis.

How to set this up depends on the OS and/or the network management software being used. On Linux, you are either using NetworkManager or SystemD, which in turn relies on either dnsmasq or systemd-resolved for DNS resolution.