Is it permissible for me to make up many missed prayers by intending to make up the Fajr prayer during the Tarawih prayer behind the Imam, so that I thereby make up four Fajr prayers a day?

There is no objection to praying Tarawih with the intention of making up Fajr, according to the view that it is permissible for someone performing a fard (obligatory) prayer to follow someone performing a supererogatory (nafl) prayer, and that there is no obligation to observe a sequence between missed prayers. However, you will not attain the virtue of Qiyam Ramadan (night prayer in Ramadan) by doing so. It is preferable that you pray Tarawih and then make up the missed prayer afterwards. You must repent and make up what you have missed daily, in a way that does not cause you hardship. Avoiding the scholarly difference of opinion regarding sequence is more cautious.