A browser window is not sufficient when a GUI requires constant live communication with the Perl program.

It is if the web service is running on the localhost. I did this years ago. I created a Dancer2 app with SQLite backend that was a full blown indoor grow operation. The front end was a browser window with a single page. All calls were made with jQuery to the webapp. The front end had several widgets that you could move around and organize, and interacted with the GPIO to turn on/off lamps, humidifiers, fans etc.

All this ran on a Raspberry Pi with a 5" screen, but I could also connect to it remotely which is a big plus.

I never did completely finish or polish the project, but it can be seen at RPi::EnvUI.