Unless you are employed/contracted directly by the school I see no reason why you should need to show your CRB at all.
If you are employed to deliver the service in schools by the council then it should have obtained/checked your CRB and schools can check with the council if they wish to.
If you are sub-contracted by a company contracted to deliver training by the council, then again that company should have seen your CRB and be able to confirm to the school if requested. This would be part of the service contracted and checked by the council.
So, why not show your CRB? IIRC, a teacher has the choice to show it to someone who asks to see it. The employer has their own copy of the CRB and isn't allowed to show it to anyone.
Why is this? IIRC, the CRB shows any convictions etc. the person has had. An instructor may, quite reasonably, not wish to broadcast to their local school teacher (who,after all, is not the person making the employment decision) that they were done for possession of drugs 5 years ago.
Of course, if 9 out of 10 instructors will show their CRB, then the tenth who won't appears weird and is likely to be assumed to have something to hide. Better then that no-one does.
Secondly, unless you are showing proper photo id (passport/driving licence) with your CRB cert how does the school know you are the person whose name is on the cert? In fact, better by far for the school to ask to see the photo id than the CRB.