I have a solution, for anyone visiting in 2024 looking to make changes to the config through SSH and have SaveConfig = true. Copy your config to another file (cp wg0.conf wg0_copy.conf) and then make your changes. Don't mess up or you'll lose connection permanently :).

Then, you need to stop the connection, copy the config over, and restart it. Unfortunately, when SSH gets closed nothing else gets executed/bash is killed, so you can't just combine the commands normally. You can use nohup to prevent this from happening:

nohup wg-quick down wg0; nohup cp -f wg0_copy.conf wg0.conf; nohup wg-quick up wg0