PageSpeed Insights doesn't support this, but you can use Lighthouse in Chrome Dev Tools to test the page locally after you have logged in.
WebPageTest offers scripting functionality where you can follow a flow to enter a username and password and load a page, or set cookies to be logged in automatically (just ensure no sensitive data is stored in the public test and expire the login/cookies immeadiately after use).
Alternatively, you could add a url param which bypasses the login for testing - for example, serves non-sensitive dummy data to simulate a logged in user. This would work for both tools.