I currently do this with Termius and ssh into the box I’m working on the launching Claude Code. Only issue I have is the occasional network issue causing the session to drop.

You likely know this, but in case you don’t: Termius makes it easy to use “mosh”, which makes your connection resistant to network drops and resumable. I am experimenting with it right now. Once you install mosh on your serve, click the “mosh” setting in the connection settings in Termius, and you are good to go.