Use the self-signed cert between the proxy and the problem device; everything else talks to the proxy.
It's not different from IPSec, GRE, VXLAN, etc. It's just the new hotness.
We use VXLAN extensively in our network, btw, and IKEv2/IPSec tunnels between sites.
Just because it can be used to recreate VPNs traditionally used to remotely dial into a secure network zone doesn't mean it shouldn't be used in far smaller use-cases. WireGuard doesn't constitute anything like a full VPN solution on its own.