I would ask really stupid-sounding questions in order to "better understand" the requirement to test two grades ahead for a one-year skip:
"Does the school normally require 4th grades to ace a 6th grade test as a precondition to promotion to 5th grade?"
"What about math? Does a child need to master Euclidean Geometry in order to get into Pre-Algebra?"
"What is the school's normal process for assessing students on material it has not introduced them to, and how often are children held back for failing to demonstrate mastery on it?"
As for the trust issue... that's why the school wants to do its own testing. They don't trust your independent results. That's common, because there is documentary evidence that private testing has often yielded a significantly higher proportion of subjects testing as gifted, leading to concerns of score inflation. The trust issue will be exacerbated in this case if your child requires accommodations, because now they're wondering how much of the results are based on the child, and how much is based on assistance (intentional or otherwise) outside of reasonable accommodations.
The upside is that if she knocks the ITBS out of the park, that eliminates a barrier between you and the school.
I'm not sure how the AIG teacher not being a good teacher influences her ability to proctor a test. There are lots of ways to be an ineffective teacher, and not all of them come into play during testing.
Because of your concerns regarding accommodations during testing, can you arrange to observe?
