A couple of random comments. Where we live SD do not accept outside testing, but I have heard (second hand, so I can't be certain is not just a rumor) that people DO use individual tests as part of an arbitration program. I think there are set dates by which families can make a case of why their Dc did not pass the group tests. Not sure, but it is worth a try to talk directly to them.
OTOH, I have heard from some people, that while those group tests DO leave out many gifted kids, the tests do identify those gifted kids who tend to do better in their programs": those that are good at following directions, both from people and in writing; are self-motivated to stay on task for long periods of time, and do not have any sort of LD. The way those programs are set up, they do not work well for children with SOME LD. However, they are a good match for students with other LD, since there is much more individualization.