My own experience with the two-step authentication is that it can be work with MFP until you don't leave your computer idle. Although it's questionable as since I trialing MFP (for 14 days) I had 3 account blockages so in other words it can be tricky.
Yesterday I used MFP for all day long without any problem while I checked my computer in every half an hour. At the evening I leave itself for two hour and when I came back the computer still running but MFP was nowhere. I thought I just closed the app. Than again an e-mail from Betfair that they take very seriously my account and as there were too many incorrect verification code entered they locked my account.
I checked the Windows event log which showed that MFP crashed 10 minutes before Betfair sent out the e-mail for me. Therefore I assume if you leave your computer and MFP idle it somehow log out and trying to log in, but now it fails because of the two step verification.
Anyway, if you turned on and use the two step verification for your account you can still use MFP. You just need to type in your username and in the password field type in your password and immediately your actual verification code. Although you do it on your own risk as there is no definite answer when your account will be locked.