A lot of worms for sure. However, perhaps one reasonable test might be whether or not the institution trains its student's to practice a specific faith. Liberty's motto of "Training Champions for Christ since 1971." seems like their objective is a religious one. If Congress isn't to "make no law respecting an establishment of religion, or prohibiting the free exercise thereof." it seems that spending tax dollars on a organization in order to train champions for Christ would be a conflict. I think a religious institution can perform non-religious services, and that those services might be subsidized by the government, however, when the service is to advance the religion, the government should not help or hinder the organization. There is little doubt in my mind that if Liberty were an Islamic institute, the call for the separation of Church and State would be vociferous.
I see. Using an institution's mission statement would seem as good a method as any to designate the net purpose of an institution which is in reality manifold. I wonder what is in fact currently the measure? Also, it would be interesting to pursue another aspect of this further: in what way is the (non-state) university receiving federal money? Are you speaking solely of student financial aid money, or are there other matters? And isn't it technically the student which receives aid? I realize that this is a technicality and a simplification which might well be described differently. But upon such technicalities is policy built. In this case, the new big question would be this: what determines the set of institutions for which students may be aided financially? An Islamic institute should surely qualify if a Christian one does. However, even within Christian colleges, there is a divide between colleges and seminaries or "Bible colleges," i.e., schools which instruct only or mainly on matters of religion and theology. Here, as in the wider question of how to differentiate between schools of varying caliber and purpose, accreditation seems perhaps the best method. But I'm well out of my depth regarding what the standards in fact are.