Definitely ask for the school's end-of-year testing for her current grade as well as for the next grade up (and maybe for the next grade up still). Sometimes only the school's own tests will help in showing teachers that there really is a need for higher curriculum. With outside testing there is just too much wiggle room for teachers to make excuses and say that their curriculum is so much more challenging than the norm and that your child couldn't possibly need more.
Also, are there emotional compenents of your DD's experience to which you can point to show that the current situation isn't good for your DD? Especially elementary school educators tend to focus on a child's emotional well being above all else (including learning!). How is your DD feeling about school in general or about herself as a learner?
Can you get a written statement from the psych about your DD and what type of educational environment the psych would recommend? If your DD is HG and possibly PG, then the standard starts-at-fourth-grade gifted program probably won't be the only accomodation that she will need. It may be a great program that she will love, but it probably won't be enough. A letter from the psych might help you establish that.
Good luck and let us know how it goes!
