Add Chat Panel to Full Screen

### Your use case

#### What would you like to do?
Add chat panel to full screen view.
#### Why would you like to do it?
It supports conversations that involve both video and messaging.
And may reduce the amount of switching between full screen and normal.
#### How would you like to achieve it?
Let the chat icon be visible in full screen, perhaps to the right of view buttons or top right (tile size may need to be adjusted for fitment).
The panel would behave as normal thereafter: It slides out from the side without exiting full screen and would be dismissible as per usual (the same as it would in normal, unmaximised view).

### Have you considered any alternatives?

_No response_

### Additional context

_No response_

Locate the full-screen view and the existing chat-panel toggle, then compare how the panel behaves in normal and full-screen modes. Confirm which view controls contain the chat icon and verify that the panel can open and dismiss without leaving full screen; test both modes manually.

Written by the indexing model from the issue text.