If it's attached through USB or registered as a HID, then yes TeamSpeak will detect and use it for hotkeys. You'd have to remove the Gamepad plugin and also select keyboard & mouse only in Tools -> Options -> Hotkeys in the bottom right dropdown.
Then after restarting TeamSpeak it is probably not be accessed / detected anymore.
Not sure why you'd want / need exclusive mode but in any case doing the above is all you can do in TeamSpeak to make it possibly not access the device anymore.
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