10.10.3 did some strange things with Google accounts. It's a step in the right direction, but it seems like there are bugs that were introduced.
You have two-factor authentication enabled; do you also use app-specific passwords? Are you using that account for mail only, or are you having OS X do other things with it? For your own security you don't have to answer those questions here. If you are using app-specific passwords, one thing worth trying would be to delete and recreate the password for Mail (or whichever applications you're using), and then update that password in the appropriate program.
That seemed to have fixed it for me. However, looking in my Console logs, I (and some others) are still getting various errors, although the operating system isn't prompting us to take action on them. Hopefully 10.10.4 will sort all of this out.