It is not valid to combine making up missed Ramadan fasts with fasting six days of Shawwal with one intention

That is not valid, because fasting the six days of Shawwal can only be done after fasting Ramadan in full. 

Shaykh Ibn ‘Uthaymeen said in Fatawa al-Siyaam (438): 

Whoever fasts the day of ‘Arafah, or the day of 'Ashura, but still owes days from Ramadan, his fast is valid, but if he intends to fast this day to make up for a missed Ramadan 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 Ramadan. With regard to fasting the six days of Shawwal, they are connected to Ramadan and can only done after making up missed Ramadan fasts. If he fasts them before making up missed Ramadan fasts he will not attain that reward, because the Prophet (peace and blessings of Allah be upon him) said: “Whoever fasts Ramadan 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 Ramadan is not regarded as having fasted Ramadan until he makes up the days he missed.