I can get them to register/sync in Bluetooth but after I disconnect them I have to redo it in the Bluetooth menu to get them to sync again.
Not sure what you mean by 'disconnect them'. I usually just shutdown RetroPie / Emulation Station and the controllers go to sleep on their own.
I use Switch Pro controllers and mine will stay synced to my RetroPie install as long as I don't plug them into the Switch to charge them. Charging through the Switch will automatically re-pair the controller to the Switch and it won't talk to anything else without a manual re-pairing.
The controller/player number lights on the controller constantly array as if it is still syncing, or as if it can't assign the player number in the controller.
I don't have a solution but my Pro controllers do the same thing so it is somewhat normal for this style controller. I think you are right that the Switch must send information back to the controller to tell it which player number it is assigned and nothing in the RetroPie package will do that at this point.
This project was for Wii controllers but maybe it holds the answer and can be adapted.