Good news! With your help, we were finally able to track down the root issue here (or one of them, at least) and deployed a fix for it this morning. Could you please try again when you get a chance? It shouldn’t time out anymore.
If you still notice any, please do let me know (especially if you get an error ID from it).
Also, if the plugin would still be helpful even without the timeouts, let me know and I can still finish that up for you.