You don't need claude code on the web for this, Cloudflare lets you spin up containers like crazy, you can boot an agent in a container, and as part of the boot process copy your claude auth token into the container. Then just ssh in, use tmux to make it persistent, and drive claude remotely.

Yeah and my preferred tools (mise) are missing from the environment, and installing it requires arcane environment configuration and then the LLM spends 10 minutes just trying to get the environment set up... On every interaction

Yeah that's why we basically built our own Claude Code Web but around Hetzner VPSs instead & terminal access. So you can use docker, open ports if you'd like. Some teams even needed us for a complicated R dev setup they wanted Claude to work with.

Check out ona.com if you haven't already. We support very similiar use cases and we support devcontainer. No limitation on language support.