I just tested it using a Wi-Fi network using a DNS server that does not block the canary domains, while my phone was connected to a VPN with a Pi-hole that does block the canary domains (I made a config file as described in the comment: iCloud Private Relay - #4 by jpgpi250 and did not use the PR). I got the message saying that the network was not compatible with Private Relay, and Safari gave errors when trying to load any webpage that Safari couldn’t connect to iCloud Private Relay.
Maybe this is an oversight by Apple, but it seems like the DNS server from the VPN is used to determine whether the relay is to be enabled or not.