How can the Hajj rituals be performed with menstruation that continues beyond the scheduled return date?

The best solution is to take medication that prevents menstruation to perform the rituals without difficulty.

If the medication is ineffective, perform all the rituals except for the circumambulation (Tawaf); do not perform Tawaf until you are pure.

If you fear missing your group, you may, according to the Hanafi school of thought, perform Tawaf while menstruating, and you will owe a sacrifice (dam).

You may enter into the state of Ihram for Hajj as a Mufrid (performing Hajj alone) or a Qiran (combining Hajj and Umrah), and you may also enter as a Mutamatti' (performing Umrah then Hajj). If you do not become pure before the Tawaf of Umrah, add the intention of Hajj to it, and you will become a Qiran. The Mutamatti' and Qiran owe a sacrifice (hady) unlike the Mufrid.