This is complicated. I see school testing and outside testing as having different but related purposes.
School testing is necessary for qualifying for services (whether gifted or IEP)-- for that reason, I'd say start that process by requesting the eval in writing ASAP. School testing concludes with educational information (not with a medical diagnosis), a team meeting to evaluate the child's needs, and a determination of whether services are necessary at school.
Outside testing gets you diagnostic information and often a fuller picture (since they will consider not just educational aspects but the whole deal). However, since you already have the ASD diagnosis, I think you may not need to start with further outside testing.
Devil's advocate, however: people who understand children with ASD may be more sympathetic testers (i.e. more able to notice where anxiety is hindering performance).
I'd start with school, but if the results are not in line with what you expect, go private for followup, making sure that private is using tests your district also recognizes as valid.
DeeDee
