It is not valid to combine making up missed Ramadhan fasts with fasting six days of Shawwal with one intention.
Is it permissible for me to fast the six days of Shawwal with the same intention as making up the days I did not fast in Ramadhan because of menstruation?
All the the Praise be to Allah.
It is not valid, because fasting the six days of Shawwal can only be done after fasting Ramadhan in full.
Shaikh Ibn ‘Uthaymeen said in Fatawa al-Siyaam (438):
Whoever fasts the Day of ‘Arafah, or the Day of ‘Ashura’, but still owes days from Ramadhan, his fast is valid, but if he intends to fast this day to make up for a missed Ramadhan fast, he will have two rewards: the reward for the day of ‘Arafah or ‘Ashura’ along with the reward for making up the missed fast. This has to do with voluntary fasts in general that are not connected to Ramadhan. With regard to fasting the six days of Shawal, they are connected to Ramadhan and can only do after making up missed Ramadhan fasts. If he fasts them before making up missed Ramadhan fasts he will not attain that reward, because the Prophet (sallallāhu 'alayhi wa sallam) said: “Whoever fasts Ramadhan then follows it with six days of Shawwal, it will be as if he fasted for a lifetime.” It is well known that whoever still owes days from Ramadhan is not regarded as having fasted Ramadhan until he makes up the days he missed.
[Via Fatwa No: 39328 Islam Q&A]