The most difficult scenario was the Knights Templar. Lost many Knights battling Cataphracts, but won the campaign eventually. Still have not figured out how to take on the Mongols. I have been mostly avoiding them in previous simulations. Fighting them at a river bridge crossing or besieging them within a city at least gives you a chance. In the open field, they are most difficult.
In reality, Abbasid v. Mongols, the Abbasid will (and did) lose. Badly. Inferior equipment, bad to non-existent tactics, terrible leadership, up against, well, the Horde, who could give modern armies lessons in discipline.
Generally speaking, yeah, your best bet against the Khan’s tactics would be to take and hold a chokepoint or beseige a city, as you’ve already found out. Failing that, a reverse Trojan Horse might work.