Private schools do get government funding here (which is a parallel to the problem).
Mmmm yes apparently it is non-denominational, but you know, non-denominational still is a Christian College by very definition of the term. You cough up the money to go to these institutions and you abide by a certain set of rules, they have strict uniform policies (which involves special -hats- even), you don't have to be of the religion to go, but it is still a Christian College, if they say to take off your non-Christian attire and stick with the dress code. Sorry you don't get any fucking sympathy from me. You could wear it at a public school, if you want that kind of freedom, go to a free school, you will still have to wear a uniform, but policy won't be nearly as strict and you are free to express any religion you want.
Private schools are just that, private institutions and how they govern themselves is going to be according to their private rules, which so long as they do not break the law (which having a dress code is not doing) then they can do whatever the shit they want. The schools are open to any religious background, don't make you convert or anything and will take anyone that can pay. If the rules and restrictions violate your degree of faith, sorry out of luck.
Asking everyone at the institution to abide by those rules is not racism of any kind. It is just rules and they are there to create the whole mystique of the private school environment, to reinforce certain constructs about power (both towards authority figures and for the students attending). This is part of what private schools are and how they work. They are not rules without a purpose. They are not rules to prevent people from attending (the bill is what that is for).
I agree anyone should be able to express their religious beliefs at a school. I also believe strongly in public education. This is not the former and so the latter part does not matter, privately owned land, privately owned standards, privately owned business.
Half the problem there is the bullshit surrounding the "better" and "Decent" education. Public school systems here in Australia aren't that bad, you get a thorough education through them, they get boned for funding (because a larger percentage of the funding goes to private schools), but they function and are just like any other school, you only get so much out of them as the student takes from it. The biggest funding difference gets diverted to extra curricular fields, sports, music and whatnot. The biggest benefit you take out of a private education is the ability to say you were privately educated and the networking that comes with that.