[Windows] Desktop pet becomes click-through and cannot be dragged

What happened

On Windows, the Codex desktop pet can become partially or completely click-through. The mascot remains visible, but clicking or dragging its body passes through to the window underneath, so the pet cannot be moved.

Steps to reproduce

1. Run Codex Desktop on Windows.
2. Enable a desktop pet.
3. Leave Codex running and use the pet normally.
4. Try to drag the pet from the center or lower part of its visible body.

Expected behavior

The full visible mascot area should remain clickable and draggable for the lifetime of the overlay.

Actual behavior

The pet remains visible, but mouse input passes through part or all of the mascot. Restarting Codex or Windows may temporarily restore dragging.

Environment

• Codex Desktop for Windows
• Windows 11
• Display scaling: 150%
• A valid custom v2 pet is installed (the issue appears related to the overlay hit region, not the sprite asset)

Additional context

This appears related to the Windows avatar overlay's mouse-passthrough or stale hit-region handling. It may be related to the existing report #34227, but this report reproduces the problem on the current Codex installation as well.

Reproduce the issue on Windows 11 with a desktop pet and 150% display scaling, then inspect the Windows avatar overlay's mouse-passthrough and hit-region handling. Verify that the full visible mascot remains clickable and draggable for the lifetime of the overlay, including after normal use and with a custom v2 pet.