Hi, Jordan. Quick update here. Thanks to your help, I was able to successfully get NextDNS running on macOS simultaneously with Private Relay by downloading the macOS app - so all good!
On iOS (and iPadOS), I still had some issues but they now appear resolved. The way I was able to get NextDNS working with a profile installed was to change the Safari Settings “Hide IP Address” option for Trackers and Websites to Trackers Only (Settings → Safari → Privacy & Security → Hide IP Address → Trackers Only). NextDNS seems not to be activated if Trackers and Websites is the active option.
I’m hoping this is a reasonable workaround on iOS. If you have any experience/thoughts, I’m open to them. Otherwise, thank you again for your quick response and help.