If you go to your profile tab, click settings > accounts > edit account, delete your password, and type in anything other than you password. Try to save this, and it will fail to log you in. Now renter your real password and it will save your account info. This acts as a “refresh” for you account token.

It’s a bit cumbersome, but there’s currently no way to manage accounts with only one logged in.

EDIT: I stand corrected, the latest TestFlight has added a logout button to the account settings page.