Try the macOS NextDNS client from the macOS app store and remove the NextDNS profile, for some reason it has stopped working on new versions of macOS.
According to Apple’s Security Whitepaper you should be able to use custom DNS settings with iCloud Private Relay:
Custom DNS settings
If a user has configured custom-encrypted DNS settings using a profile or an
app, the DNS server specified will be used instead of ODoH. Safari connections
and all unencrypted HTTP connections will also resolve names using the
specified DNS server prior to routing through Private Relay.
An unencrypted DNS server provided by a local network or manually edited
in Settings (iOS) or System Preferences (macOS) will not be used for iCloud
Private Relay traffic.
Source: iCloud Private Relay security – Apple Support (AU)
From my personal testing you can use NextDNS with iCloud Private Relay on iOS. Your devices IP address should show up as that of iCloud Private Relay IP address in your NextDNS logs.
Make sure you follow the intructions on the NextDNS website and install the configuration profile from apple.nextdns.io on your iOS device.
Make sure to update your device to the latest version of iOS and if that doesn’t help I would recommend resetting your network settings:
Settings -> General -> Transfer or Reset iPhone/iPad -> Reset -> Reset Network Settings