Bazzite: No Steam Overlay in Gamescope Gamemode

Nobody has claimed this yet.

### Is there an existing issue for this?

- [x] I have searched the existing issues

### Is your issue described in the documentation?

- [x] I have read the documentation

### Is your issue present in the latest beta/pre-release?

None

### Describe the Bug

**Main Issue**:
- When running Sunshine during Bazzite's Gamemode (Gamescope) we are unable to get the Steam Overlay UI to show.
- You will hear a sound, and it will show on the native screen, but it does not show when streaming.
- https://docs.bazzite.gg/Handheld_and_HTPC_edition/Steam_Gaming_Mode/

**Other Minor Issues**:
- These issues could be completely unrelated, but I included if it helps with troubleshooting. More details in the links below.
- No mouse pointer. Using traditional methods of troubleshooting fail. Have directly connected mouse.
- Image distortion in normal desktop mode.
- No systemtray icon, but this can be explained with the fact the gamescope gamemode does not have a desktop. Restarting the Sunshine service in desktop mode will show icon. This is a cosmetic bug only.

### Expected Behavior

Steam Overlay show be shown when using a directly connected controller or using the 'Home/Guide Button Emulation Timeout' feature.

### Additional Context

**Clients Tested:**
- macOS Native: 6.1.0
- Linux Flatpak: 6.1.0

**Related:**
- https://github.com/ublue-os/bazzite/issues/1276
- https://github.com/LizardByte/Sunshine/issues/3556

**Troubleshooting**
- Tried multiple resolutions
- Manually set monitor and gpu in config
- Enabled: Strictly enforce frame bitrate limits for H.264/HEVC on AMD GPUs
- Some users report the overlay being shown with HDR enabled
- When switching to mouse emulation mode I can briefly see the overlay

**Additional Comments**:
- I am not sure if the Sunshine team can help on this one, but I figure I post this over here as well to see if you had any insight.
- Another sign pointing to a larger issue is this is it can be replicated with SteamLink as well.

**Full Log File:**
[sunshine.log](https://github.com/user-attachments/files/20102794/sunshine.log)

**Screenshots:**
_Local:_
_Remote:_

### Host Operating System

Linux

### Operating System Version

Bazzite 42 - Linux 6.14.4-103.bazzite.fc42.x86_64

### Architecture

amd64/x86_64

### Sunshine commit or version

v2025.501.153943

### Package

Linux - Fedora Copr

### GPU Type

AMD

### GPU Model

AMD RX 9070

### GPU Driver/Mesa Version

Mesa 25.0.5

### Capture Method

None

### Config

```Shell
min_log_level = 1
sunshine_name = GamingPC-AMD
```

### Apps

```json

```

### Relevant log output

```shell
[2025-05-08 07:29:37.209]: Error: Couldn't load cuda: -1
[2025-05-08 07:29:37.211]: Error: Environment variable WAYLAND_DISPLAY has not been defined
[2025-05-08 07:29:37.334]: Error: Couldn't find monitor [0]
[2025-05-08 07:29:37.335]: Error: Couldn't find monitor [0]
[2025-05-08 07:29:37.537]: Error: Couldn't find monitor [0]
[2025-05-08 07:29:37.537]: Error: Couldn't find monitor [0]

[2025-05-08 07:29:37.208]: Warning: Failed to create system tray
[2025-05-08 07:29:37.868]: Warning: [av1_vaapi @ 0x3db44000] Multiple slices were requested but this codec does not support controlling slices.
[2025-05-08 07:29:37.949]: Warning: [av1_vaapi @ 0x3da38fc0] Multiple slices were requested but this codec does not support controlling slices.
```

Start with the attached Sunshine log and reproduce the issue in Bazzite's Gamescope Gamemode using both Sunshine and SteamLink, then compare the related issues #1276 and Sunshine #3556. Done means identifying a project-side cause or a clearly documented external dependency, with the Steam Overlay rendered in the remote stream without regressing the reported desktop-mode behavior.

Written by the indexing model from the issue text.