Thanks for your post and welcome to the TeamViewer Community.
That is a really good and interesting question! But this part of TeamViewer cannot be customized. I think you already know that the TeamViewer QuickSupport and the Host module can be customized, but when the connection is established you will see again a TeamViewer icon in the little panel on the right side.
Please feel free to post this customization as an idea into our ideas board to let our Product Management know, that you would like to be able to customize the panel as well.