I've built something similar to run in my own cloud to run my own container images and add some features. The Claude Code web app has been pretty buggy for me and I like to run these things myself. If I do run into bugs or issues I still have access to the containers (and I do, because I vibe coded the whole thing). I gave it service accounts to the cloud it's running in, where the CI builds take place and setup dev environments per PR that deploy automatically. This is a massive productivity boost imo.