Well its possible for Byzantine to win against the doomstack if they put all their troops + holy orders in Constantiople and just wait for the horde to arrive.
The Ilkhanate lost all its troops in a war against the byzantines.
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The problem is neither overpowered BYZ nor underpowered Mongols. The problem is that the Mongol AI doesn't seem to be willing to surrender, no matter what happens. This means they'll grind their troops until they lose all of them and the enemy forces the surrender after getting them to 100% war score. A 100% war score probably won't happen until a fresh Mongol horde goes down from 120K troops to 40K. A bloodied horde might go from 90K to 20K or something.
More about grinding in a minute:
Now that the Mongols no longer appear as a 120K monolith stack but instead use ~20K stacks that may or may not gang up, they are capable of funnelling their troops through a bottleneck. If the terrain is difficult and takes long to travel, it's possible that a high-quality 30-40K defending stack (e.g. cataphracts, holy orders) could keep killing the Mongols as they come, 20K after 20K after 20K, so that the Mongols at no single time outnumber the defenders in the field.
I would know. I've tackled both the Ilkhanate and the GH in this way as an Italian republic with holdings in Cumania...
Put a cataphract retinue and/or holy orders and/or mercs in a well-developed province with mountains and/or a river or two rivers, and that just might happen.For example, if the AI Basileus managed to farm a huge cataphract retinue (after teching in military organization) and park it somewhere mountaineous and river-y in Armenia or Georgia — and keep reinforcing it with mercs (e.g. a 15K band) and even some serious levies, then it's possible that the Ilkhanate fed itself into a bottleneck gradually.
And if the Byzzies were parked in Derbent or somewhere like that, then it's possible the Ilkhanate went around the Caspian Sea, which promotes bottlenecking.
Bottom line, the Mongols will be defeated if a large high-quality defending stack (preferably with terrain bonuses) is never outnumbered, i.e. manages to break the morale and thin the numbers of subsequent incoming Mongol waves for long enough to just never really be outnumbered in the battle. The outcome would be different if the CK2 combat system simulated troop fatigue from fighting.
I do not think the Illkhanate had 120k troops it was more like 100k when they attacked the Byzantines and the ERE had 110k tropps according to the ledger now they have 90k in a time of peace and I do not know why they increase their teritory by conquering baghdad.
I have no idea when that was but with all of Greece, Anatolia, Serbia, Bulgaria, Jerusalem, Armenia, Georgia, Crimea, Syria and most of Arabia I think they can gather that much.
Current 'levies' include retinues, mercs, holy orders etc., so they can be pretty huge and exceed the total levies figure, so it might be current 110K and total 50K, for instance.