I don't know about a diagnosis of SI but the following author is one of my favorites on the topic. Easy reads.
http://www.out-of-sync-child.com/
This is a great book about SI and if you read about SI and think that your child could have it, you should definitely check it out. An evaluation by OT is a good idea if you read the book and feel that your child might have it. It can be a comorbidity though so it might not be the only issue you need to deal with.
As far as out of pocket educational testing goes, sometimes if you look around at those private tutoring agencies, they do offer free testing with assumption that you will be enrolling in their services. The age that they offer the testing might be the determining factor for whether you can do this.
Being seen by a developmental specialist is a great idea, particularly if you live close to a large children's hospital that will have all the services you might need. Your pediatrician will likely be happy to refer you if you ask. However, here's the clinker. Most insurances do not cover "developmental issues". I know this as I have referred patients for similar issues and parents have come back telling me the insurance company won't approve the referral. Autism, or the spectre of it since it is in the press so much, however, does seem to get things approved most of the time so if you do go the development route, ask you pediatrician if they think your child could be on the autism spectre and use that is your basis for referral.
Lots of people have already posted great advice and I hope things work out for you and your child. Good luck!
