Copilot Chat Profile selector CSS issues

[NakosPTI]

Describe the bug
Copilot's agent selector CSS is broken.
On dark theme it has z-layer issues, and is hidden from view.
On light theme, it is shown that besides the previous issue, CSS is not applied to it.

Also, on both themes, the dropdown can shift the layout of other elements.

The cause of this might be that the chat profile selector dropdown is added to the main DOM instead of the the shadow-root.

To Reproduce
Based on chainlit/cookbook/copilot:
https://github.com/NakosPTI/chainlit-copilot-agent-select

Changes from chainlit/cookbook/copilot:

• Changed port to 801 due to CORS issues.
• Added light theme to the embed configuration.

Expected behavior
Chat profile selector should work the same as it does on the default chainlit frontend.

Screenshots

Desktop (please complete the following information):

• OS: Windows 10 22H2
• Browser
  • Firefox 135.0.1 x64
  • Edge 133.0.3065.69 x64
• Python 3.11.9

Additional context
Issue was observed with chainlit 2.2.0 and 2.2.1.