If private testing is prohibitive, then you can use an achievement test like the SAT and a portfolio.
She is 6 and the low cost test like ACT/SAT or EXPLORE will not be a good fit right now :-).
I guess that she is in 1st grade now. How's her reading and Math level? Is she doing at least > 2 grades above (3rd or 4th, etc)? What's her interest? Is the school not being challenged enough?
If the school is a good fit, I would wait until she gets a little older to be able to take like EXPLORE (that costs <$40) through talent search. (EXPLORE may go away for good in a few years but hopefully there will be a new test (for 3rd to 6th grade level) before the kids are old enough to take SAT/ACT).
If she is reading at middle school level and doing multiple digit multiplication, I would encourage for you to talk to the school and the school district to see if they can give IQ test or achievement test for your grand daughter.
And you can submit it along with a portfolio (the video of her talking/explaining something she is passionate about whether it be a book or scince stuff, her math work sample (at least 2 grades above her age peers, etc....).
Good luck!