The first blanket GT testing in our school district is in 3rd grade, and I think it's the WJ, so it is possible that "real" achievement testing is indeed what they mean. But definitely ask! Different districts are different.
The rest of this post assumes that the school will give a YSP-approved test. If not, then I'd recommend paying for it now on your own.
But if the school's achievement test is okay to submit to YSP, then I'd say that I think testing can be useful at any age, given the right test, the right tester and the right questions that you want the test to answer. It depends on what you're looking for. Given that you want to get YSP-level scores, I'd actually think earlier would be better than later.
Aside from possibly getting the test for free, paid for by the school (as we did), I don't really see a reason to wait based on what you've told us. I don't think either choice is a bad one though. Are you patient enough to wait?
Oh, and I'd check to be sure the older test scores that you've already submitted won't expire. You don't want to wait, only to find that now you have to get the IQ testing done again!