Squeezelite should not be stopped by the speaker. The bluetooth output should remain active on one Squeezelite instance and the speaker should be able to reconnect when started. You can check the Bluetooth menu for the entry which should indicate trusted and paired and connected when active.
Best backup option to get the whole system safed is to create an .img file with our image burner plugin, which you can burn or save onto another device. However, saving the LMS settings should also not be an issue. We recently upped the max file size because someone had a huge save-file, how big was yours? This might the issue here.