Yes, I experimented a bit with various devices, and at least in my case, the web app now loads when I specifically "Turn off Content Blockers" for the IP address where CARP is running. The content blocker I'm using on my iPhone and iPad is AdGuard. There's probably something inside AdGuard's whitelisting settings I can tweak to figure out which component specifically is causing these hiccups, but I haven't gotten that far yet in the troubleshooting. I just configured the webpage settings so that content blocking is permanently disabled, for that one site.
As mentioned though, it works fine on my Mac -- where AdGuard is also installed and running -- without having to turn off Content Blocking. And that's the screen where I'm using CARP primarily, anyway.