Hi there, @alamb19 ...
In my own testing of the Studio Capture tool within Canvas Studio, I could re-create a similar scenario as you've described. In order to remedy this, I would recommend that you have an additional browser window open with any websites (which may include you logging in to your Canvas course again) that you would like to show your students. That way, when you start up the Studio Capture tool, you can select the new window you opened to show whatever you want to your students.
I know this is a bit of a work-around, but I think it would work for your needs. Keep us posted here in the Community if you have any other questions.