If you are testing because you believe your child is gifted, and *especially* if you think your child might be HG+ and/or 2E, it is extremely important to use an evaluator who has a lot of experience with gifted/HG+/2E children. And don't take the evaluator's word for it; ask around if at all possible and find out about other families' experiences.
IMO, it is well worth it to get a good evaluation the first time and have to pay more than get a bad evaluation for less money, and not get the answers you are looking for.
Maybe I missed something, but you didn't mention how old your child is, what testing you are considering having done and the reason you want the testing done... depending on the answers to those questions you will likely get difference responses.
At the extreme of reasons you have some parents that want to test simply because they want to reassure themselves that their child is special... Those parents will often shop their kids around to be tested over and over until they get a score that they believe fits their expectation, might sound crazy but it happens.
Other parents have noticed things about their child and want to make sure everything is fine and to see if he/she could use a more advanced learning environment than they currently have... No one knows what reason you have for looking at having your child tested... hopefully it is for the right reasons and not the wrong ones.
As for your concern that a psychologist is going to give you a better result than a grad student... Maybe yes, maybe no... some psychologists don't spend as much time or effort in testing as they should, some grad students take it more seriously than psychologists... your child might also be impacted by who the tester is.. they might respond better to a male than a female, maybe it is reversed... maybe they respond better to someone that looks younger like the grad student or maybe better to the older person that reminds them of a grandparent.. there are a lot of factors and none of the answers you get here are going to be able to factor in information about how your child interacts as you will be able to.
As for the type of test you are having done, if it is for placement in a gifted program then you need to find out the tests that are acceptable, if it is just for your own piece of mind then there are other less expensive tests that will placate your fears but likely wouldn't be accepted for placement, but again I have no idea what your reason for looking into testing is.
Without knowing your reasons for testing it's hard to say whether or not it's important to have a licensed psychologist do the assessment, rather than a grad student. I assume you're looking solely at educational (GATE) testing? If your local university offers the assessment then one would think the grad students there are perfectly capable, with supervision, of handling an assessment. If you believe your child has other "issues" (2E) it may be advisable to have someone with more experience handle the testing.
Personally, we never considered an assessment of intelligence until it became clear that it was necessary for our son. His behavior in school was becoming an issue. His teachers recommended an assessment and we chose to go to a specialist in gifted (GATE) and 2E testing. The doctor was wonderful with our son, who thought he was going to "play games" that day. He went above and beyond with the testing we requested when he realized that our son was extremely precocious. If you have the money, I highly recommend having someone experienced in gifted educational assessment do your testing. If you're strapped for cash, I'm sure the university will provide a perfectly, appropriate testing experience.
I think I've probably put you right back where you were...sorry! LOL
Best of luck to you!