See, I agree with your initial stance, I disagree with the later.
The arbitrary thing becomes questionable. For example, with regards to the Christian one, is it simply what we define ourselves as to what others must accept us as? For example if its a Christian group that believes those who are homosexual can not be "christians" because they are routinely and knowingly, without any attempt to repent or change the fact, engaging in a "sinful" life style? Should they be forced to allow someone they don't believe is Christian but considers themselves such into their club to be able to get funding? And if so, who makes the determination of what's "arbitrary" or not? In a similar vein, if an "African American" club restricts its membership due to race does it have a right to deny someone who wants to join the club because the actions being undertaken are important to him due to being raised by a black step father, but the club doesn't consider him African American so would deny him entry?
It seems to me your later statement is simply interjecting an additional way for discrimination to be allowable if its in such a way that doesn't bother the college in an ideological sense, again creating a defacto situation here they can discriminate against people with public funds based on arbitrary political views.