Don't do that. You're jumping too high at this point. REmember, at this point, they think that they never notified you, you called for notification, they told you. You need to be sure that they know that that ain't the case. Start with a letter to the Dean of Admissions. What you say will depend on what you want, but you need to share your side of the story.
Tell them again when you applied, and that you didn't hear from them. You finally contacted them yourself only to find they thought they were missing your GMAT. Explain how and when you submitted this data. (Showing that they're wrong.)
Next, talk about how when you drew it to their attention, you suddenly received a ding. It's your feeling that they did not consider your app until after you made contact, and you feel it's unlikley your app got the full consideration accorded to the other students who also applied on time with all necessary info. In fact, you recognize that the class was already probably full by the time you contacted them.
You recognize that sh!t happens, but are disappointed that the work you put into the application, as well as the fee, did not receive the same consideration as everyone else. You would like __________ from them. (Consideration of same app next year? Refund of fee? Free application next year? Meeting with adcom to discuss your application?)
In closing, this experience has not lowered the esteem you feel for this school, and you would still very much like to pursue your education there.
~OR~
In closing, this episode has adversely affected your respect for this school. While you understand that mistakes happen, you are particularly offended by the way this was handled once you drew it to their attention. You sincerely hope they put procedures in place to prevent such things from happening to other students. And they sure as hell won't be seeing you next year.
That's my take.