I think you definitely have the credentials to land an internship with one of the Big 4. The most important things they care about are someone being intelligent, someone having leadership experience, and someone they can put in front of a client. From what you have written, it sounds like you have all of the above - you would most likely pass your interviews.
The challenge comes with landing one, since you say they do not look favorably upon your university. Do they recruit at your university at all? Do they host on-campus interviews? Usually that is the best way to go. You don't need to know anyone in the firm at all, especially if you have a great GPA and resume, which it sounds like you do, and if you write an awesome cover letter. They will speak for themselves. The only problem with the Big 4 is that their summer internship recruitment is over. I would say see if you can get in touch with the main recruiters for your area and ask them. You can also always apply on the Big 4's websites.