What you are looking for, actually, is post-doctoral fellows doing their year of APA-required clinical internship in an on-campus clinic. Cheaper than a board-certified psych, definitely under the supervision of a licensed psych, and likely to be more skilled than a pre-doctoral grad student.
This is what DS's testing entailed at a major university. We had an actual neuropsych looking through his records and he spent about 30-45 min. with DS but the rest of the testing was done by a fellow. There was also a student who interviewed me and filled out parent inventories while testing was done on DS. It wasn't to test for giftedness though and this particular psych sees only specific types of cases--in DS's case he had a head injury and our insurance paid for all the testing. I don't think he would see someone just to give an IQ test and I'm not sure that the fellows would either. But maybe there are other universities that do this.
