Honest suggestion - ask the agent to figure a compat shim out, the files are jsonl stored at your ~/.claude/sessions you can most likely just reshape it to work on OpenCode or similar, or have a different Claude Code config that points to OpenRouter or other API style endpoint CC supports and then you can swap accounts and it should still work!
I don't use any other harness, but I have a cron that picks up changes in my jsonl every X minutes and writes them to a SQLite database with full text search. I also have instructions in my user level claude.md (applies to all projects) to query that database when I'm asking about previous sessions. That's my primary use case where I want it to grab some specific details from a previous session. I have terrible context discipline and have built some tools to help me recover from just continuing a different task/conversation with the wrong context.
I could search it myself, but haven't needed to. Getting it out of SQLite into some format Cursor understands should be trivial.
I'm trying that out with Cursor now. But it does take some work to get it to the same state with subagents and making sure it understands the state of the progress that was interupted. But it seems worth the time to get a solid skill defined up and running that can do this, given that's it's an almost daily event by now.
Maybe a good candidate for a Claude Routine! "By this time each day, brace for upcoming outage by preparing a comprehensive information package for Cursor to take over your work on active sessions" ...