Thinking of They-Who-Must-Not-Be-Named, I realized the interest of Mongols would be incidentally, ITTL, be drawn to the Near East way early.
That's because of Georgia.
IOTL, they came into collision with Georgia barely after chasing down the Khwarezmians into Azerbaijan, resulting in 1222 in the
battle of Khunan. ITTL, Georgia is not so isolated as the strong Byzantine empire here borders it. And given the standing alliance between these two, I'd expect the Byzantines to provide some assistance, even if it's too late.
The battle of Khunan was just an incursion, a big reconnaissance, so if any further clashes happen, the result would not be significant until the Mongols really commit to it.
In worst case scenario, this could draw Gengis Khan or Ogodei ( Gengis died IOTL only in 1227 and his succesion by Ogodei was rather smooth) into invading the Empire by the late 1220s or early 1230s (they already hold onto Persian highlands and Kalka river gave them virtual domination of the Ukrainian steppes, so they can launch a two pronged attacks into Anatolia and Balkans).
This somewhat reminds me of the 'knight school's established by William de Tancerville at the Chateau de Tancerville. It was technically simply Tancetrille's household, but it acted as a place for other nobles to send their younger sons to gain military and social training.
Might we see a local noble or Christian Knightly Order established a similar establishment (or,.more than one, for that matter). The frontier state of the Crusader Kingdoms means that having a place to train knights and soldiers would be good for the state, and it might also act as a place for locals to enter, win a name for themselves, and hopefully land in the service of the Prince or other nobles.
These retinues are tied to the persona of the lord, and not to an institution. As such, they rely on that persona to exist and wouldn't survive them.
As for the knightly order, that's by definition an order, where training is a mean to an end and not an end in itself, the service in that order being that end; you are guanranteed a training, but that's part of your engagement into the order and your service here, not some school lessons on tactics, strategy or physical training you can get just for the goal of getting trained. So in my mind, the concept of this evolving into a military school of sorts is not fitting the mold. Plus, there is no underlying trend and circumstances that justify, beyond 'divine author intervention', making this going true. There is some reason it took until the 18th century to happen, and it's not after 45 years of butterflies, or event a century, that we are going to change that.
EDIT : However, there was already an oncoming if not ongoing pattern of establishing medical and law schools across the Mediterranean in this time. What can be considered is whether or not accelerating this trend here.