I do wonder if it was possible for the Song to outlast the Mongols. The root of the First Mongol-Song War seems to be that the Song attempted to seize several of their old cities from the Mongols without realizing that said cities had lost their importance and value. Further peace delegations from the Mongols were initially held up and executed by a local governor.
During the full swing of events the Mongols had a extremely hard time fighting the Song who used geography and unparalleled military technology to beat back the Mongols. Importantly at this time period the state of the Empire was in flux due to the death of Great Khan Mongke and rivalry amongst the factions. The Song may have been able to play factions against one another.
Further it is possible the Mongols could have taken the same policy of the Liao and Jin dynasties and exacted a much more fruitful tribute from the Song.
During the full swing of events the Mongols had a extremely hard time fighting the Song who used geography and unparalleled military technology to beat back the Mongols. Importantly at this time period the state of the Empire was in flux due to the death of Great Khan Mongke and rivalry amongst the factions. The Song may have been able to play factions against one another.
Further it is possible the Mongols could have taken the same policy of the Liao and Jin dynasties and exacted a much more fruitful tribute from the Song.