Thanks a lot for the help, Alex.
If you wait 60 minutes, the access will expire and you'll be able to refresh. I know it's not ideal for testing, but this is the only way I'm aware of. There's also some discussion about this in the Zoom forums: Error in Refreshing with refresh tokens - #6 by tommy - API and Webhooks - Zoom Developer Forum
I hadn't considered JWT because the guide said you don't allow it and I saw a few threads in the forum saying the same: How to Refresh JWT Token with Custom Auth - #6 by yogi, JWT Authentication of REST API - #2 by byron, Display HTTP GET result - #7 by alper
I'll give it another try and let you know what I find.