From the ISEB site:
Can pupils from state schools sit Common Entrance?
Yes, pupils from the maintained sector can sit Common Entrance and many do, particularly at 11+. Pupils are usually entered by their parents using the registration form available in the parents' section of this website. At 13+, advice should be sought from the chosen senior school about the papers which need to be taken, and at which level. Some pupils from the state sector will be able to sit the examinations in their current school; others will need to find a suitable sitting centre, such as a local prep school or a tutorial college.
Probably more difficult to do from state school as they will be doing other things at the time whereas state schools will be full speed at the CE.
Bullying is intolerable and if he hates the school then move him - if he hates it so much then he won't do himself credit in the CE