Following the provided information, I suppose that you already get know the following KB article: How to Select and Copy Text in Chat Messages - Telerik UI for WinForms. Should you have missed it, I would highly recommend referring to it as the solution I propose here jyst extend the one in the above-mentioned KB article.
In order to copy text using shortcut (Ctrl+C) you should use the same approach for displaying RadTextBoxControlElement in the ChatMessageBubbleElement. However, the input text box in RadChat control steals the focus every time when RadChat control is focused. To prevent this, you should create a custom chat control which inherits from RadChat and override the OnGotFocus method in the following way:
Thus, you should be able to copy text from the RadTextBoxControlElement using shortcut and paste it whenever you like. Take a look at the attached gif file.
I hope this information helps. If you have any other questions, please let me know.
Regards,
Nadya | Tech Support Engineer
Progress Telerik