When you use TeamViewer to connect to a remote computer, with Windows Authentication, TeamViewer QS will restart on the remote computer with those credentials.

Except it was throwing a Microsoft UAC prompt, asking for those Admin credentials, which I’d already entered.

Except it’s not supposed to do that according to TeamViewer.
(TeamViewer: Is TeamViewer UAC Compatible)

The fix?
Under this Group Policy setting,
User Account Control: Behavior of the elevation prompt for administrators in Admin Approval Mode
we were setting it to “Prompt for credentials”

Changing that to Elevate without prompting resolved that.

Isn’t that a security problem?
Yes and no.
Yes, and I wouldn’t be actually changing that Group Policy setting
No.  What I’d probably do is change the underlying registry key to 0, connect to the client computer, and then change it back to 3 when I was finished.

Further reading:
User Account Control Group Policy and registry key settings
Remote assistance not allowing UAC prompt interaction with the box checked to allow
How User Account Control works (includes information on what each of the different coloured UAC prompts mean)