Can I fast the six days of Shawwal before making up missed obligatory Ramadan fasts in the Hanafi school?

So in the Hanbali Madhhab as you asked, yes you can, you can do the six days of Shawwal and then make up your menstrual fast afterwards. That's actually the view of the majority of scholars as well. In the Hanbali Madhhab specifically, you need to have done your missed days of Ramadan before the days of Shawwal because the Prophet said "man saama Ramadan thumma atiba'ahu" whoever fasts Ramadan and then after that completes the six days of Shawwal. So they say you must do your Ramadan fast first. The majority of scholars say no because Aisha for example used to make up her Ramadan fast throughout the year until or before the following Ramadan and she would have of course been amongst those who were doing the six days of Shawwal as well. So there are two views on this. The safer view is if you are able to do your Ramadan fast first then do that but if not then according to the majority of scholars you would be able to do your six days of Shawwal and then do the Ramadan fast after if you were taking that particular view.