Is it permissible for me to fast the six days of Shawwaal with the same intention as making up the days I did not fast in Ramadaan because of menstruation?.
That is not valid, because fasting the six days of Shawwaal can only be done after fasting Ramadaan in full.
Shaykh Ibn ‘Uthaymeen said in Fataawa al-Siyaam (438):
Whoever fasts the day of ‘Arafah, or the day of ‘Ashoora’, but still owes days from Ramadaan, his fast is valid, but if he intends to fast this day to make up for a missed Ramadaan fast, he will have two rewards: the reward for the day of ‘Arafah or ‘Ashoora’ along with the reward for making up the missed fast. This has to do with voluntary fasts in general that are not connected to Ramadaan. With regard to fasting the six days of Shawwaal, they are connected to Ramadaan and can only done after making up missed Ramadaan fasts. If he fasts them before making up missed Ramadaan fasts he will not attain that reward, because the Prophet (peace and blessings of Allaah be upon him) said: “Whoever fasts Ramadaan then follows it with six days of Shawwaal, it will be as if he fasted for a lifetime.” It is well known that whoever still owes days from Ramadaan is not regarded as having fasted Ramadaan until he makes up the days he missed.