Steam device: Bluetooth gamepad becomes unresponsive when launching and/or when you end the streaming

Describe the bug
Bluetooth gamepad (xbox one bluetooth gamepad) becomes unresponsive as soon as moonlight is launched and/or when you end the streaming with LB+RB+Select+Start button combo

Steps to reproduce
Launch moonlight in a steam device, can't interact with moonlight interface
Turn off and then on the bluetooth gamepad, now it works and can interact with the moonlight interface
Start/Launch a game/stream
End gamestream with LB+RB+Select+Start button combo
Gamepad is unresponsive again
Turn off and then on the bluetooth gamepad, and it works again

Workaround
Turning off the gamepad and on again makes it work again.

Screenshots
N/A

Affected games
N/A

Other Moonlight clients
It doesn't affect my phone's android client

Moonlight settings (please complete the following information)

• Have any settings been adjusted from defaults?
• no

Gamepad-related issues (please complete if problem is gamepad-related)

• Do you have any gamepads connected to your host PC directly?
• An xbox 360 wireless dongle and wirless gamepad, dongle is always connected/on, gamepad was not on.

Client PC details (please complete the following information)
Steam link device

Server PC details (please complete the following information)

• OS: Windows 11 Pro 24H2, 26100.4652
• Sunshine Version v2025.628.4510
• Moonlight Version: v6.1.0
• GPU: Gigabyte Nvidia RTX3060

Start by reproducing the issue on a Steam Link device with an Xbox One Bluetooth gamepad, following the launch, stream-start, and LB+RB+Select+Start end-stream steps. Compare behavior before and after reconnecting the gamepad; done means the gamepad remains responsive through Moonlight launch and stream termination.