In case you can not get a real answer (as I have never seen anybody post about T'way, whoever they might be), you might try contacting them and asking:

. Do they have a transit desk at ICN

. Will they let you check-in at the gate? [And if so, can you get through transit security w/o a BP, I forget]

I am always a bit nervous about relying on on-line check-in for such airlines. And 2 hours could be dicey to go groundside.