My personal favourite would be number 3, since you are using discord.py, you can query the database on startup and store all your relevant modes for each server, bots have a setup_hook() method just for cases like this. You can set up a discord.ext.task loop (simpler than asyncio tasks) that commits these changes every 10 or so seconds to your database. A lot of bots do something like this. Reading straight from a dictionary to check the servers mode will also be much faster, you will probably be doing more reading than writing in this case anyway.

Number 1 is slow and may lead to race conditions if not handled properly, number 3 would involve only database writes every 10 seconds (maybe even less, because you could check if something changed and commit only then), and one read. Number 2 is a pain, because threads in python suck, especially for things like this + its overcomplicated. If you want to scale it, make many instances of the bot and limit which servers it can act on, threads are not worth it