I've just seen your post, and I kind of went through similar pain, but went the other way round: a small self-hosted hub owns the Claude Code sessions, and the client is just Telegram, Slack or a web page, so a reconnect never touches a session. It does need a claude.ai login, though, same as Remote Control. Open source, in case you think it would help: https://github.com/enes-alatas/spool. I'd love to hear some feedback as well.