I just wanted to give a quick update. The school agreed to test, so we filled out all the paperwork the week or so after school started in January. He's about halfway through now, and just loves all the tests.
Meanwhile, other teachers came up with the idea of allowing E to read in class, before he starts being disruptive. It was the same idea I suggested months ago, but I guess it was more palatable with other teachers suggesting it. Since this started, E has not been sent to the office because of disruptive behavior in the classroom. Now his teacher complains that he isn't getting work done in class.
(He is still being sent to the office for misbehavior during recess, but that's probably another post.)
Again, many thanks to everyone who weighed in and offered advice and opinions!