If they don’t require non-Christian religious charities to repudiate their faith as a condition of government largess, then discriminating against Christianity clearly and unquestionably violates the establishment clause.
Quite frankly, I would have nothing to do with a "charity" that relies on government largesse for its financial well-being. That's exactly what got Catholic Charities into trouble in California a few years ago when they tried to claim some kind of exemption from a law relating to domestic partnerships, on the grounds that they could not be compelled to violate their religious beliefs. The state judge in the case rightly decided that Catholic Charities isn't a religious organization, and one of the key points in his ruling was that they received most of their funding from government agencies.
It's instructive to note that in the parable of the Good Samaritan, the Samaritan who came across the injured traveler didn't pick up a cell phone and call 911 so a chariot could come and take the guy to the King Herod Trauma Unit at the Caesar Augustus Medical Center. The Samaritan put the guy on his donkey, took him to a nearby inn where he could rest and recuperate, and paid the innkeeper out of his own pocket.