Using your screenshot there, return to that area and try clearing all the Google cookies. Either using Manage Data to search for them & erase them that way. Or using the Clear Data button to clear them all at once. You can also clear your cache from there.

Hopefully clearing all those things will help Gmail create a cookie that will remember you much longer & not always bug you for authentication. That is one of the reasons why I hate 2 factor authentication, when it acts up & always asks you to enter a code just to sign in.