You would need to override the ControlTemplate. You can see some other suggestions in the official feedback portal item here: Chat: Expose Input Area Template.
n my comment's GitHub link
, you could add items around the ChatEntry control in the ControlTemplate, but not edit the ChatEnty control itself (other than inherited features from Entry).
This will probably get you close to what you're looking for you, however, that's as far down as you can go for the overall template at this time.
Here's a screenshot to help guide you:
Lance | Technical Support Engineer, Principal
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