AWS keeps signing me out when I try to look at my VPC from different regions outside of the U.S.

Switching regions within the console really shouldn't expire the logged-on session. It could be some settings on your browser. Have you tried using a different browser or from a difference machine? Also have you checked if there is any SCP policy in place that is denying your access to other regions? Is it this only happening for VPC or all other services when you switch regions?