Steam Deck LCD - Google Stadia controller not correctly reflecting battery percentage in icon

Your system information

• Steam client version: 1782861641
• SteamOS version: 3.8.14 (OS Build: 202660703.1)
• Opted into Steam client beta?: No
• Opted into SteamOS beta?: No
• Have you checked for updates in Settings > System?: Yes

Please describe your issue in as much detail as possible:

When connecting a controller I except the icon to go down (the green part) in the bar so I can see how much charge is available.
Currently the charge in the icon is always 100%.

Steps for reproducing this issue:

1. Connect a Bluetooth mode Stadia Controller to the Steam Deck LCD.
2. Check the top right where there is a second battery icon with a controller next to it. It has 100% charge as seen in the icon.
3. Open quick settings, go to Bluetooth and suddenly see that there is only 15% remaining in the controller.

It would be nice if the actual percentage would be the same as in the internal Steam Deck battery. I am also not sure if this is a bug or not.

Secondly, not related to the issue, but just checking if someone knows, the quick settings button on the Stadia Controller doesn't exist, but I read that for some people the Assistant button is working like that.

I tried lots of things to map it, to no avail. And searching online is also not giving any results. So for now when connecting to a TV, I need to get up, and go to the TV to open the quick settings,
would be nice if that could be mapped/map able (it currently is not possible as far as I tried things).

No files, tests, or entry points are mentioned. Start by reproducing the Bluetooth Stadia Controller connection on a Steam Deck LCD and compare the top-right controller battery icon with the percentage shown in Bluetooth quick settings. Done means the icon reflects the reported battery level; clarify separately whether the Assistant-button mapping request is in scope.