A There is no legal reason why you should not co-opt your school keeper. Your chair is showing healthy caution, but tell him the law is pretty hot on what you can't do but often unhelpful on what you can. In fact under the new School Standards and Framework Act 1998 all schools except the very smallest will be obliged to have an elected member of the support staff on the governing body from next September.
You also ask what I think. I can only say, not very helpfully, that the answer will vary with circumstances. The governing body I belong to has for many years co-opted an extra teacher (the next in line in the voting) and a support staff member, because a majority of us have believed this gives a better balance. We do, however, like the support staff to choose a representative to nominate for co-option.
Balance is something about which every governing body must make up its own mind. It varies from one school to another. If, for instance, extra teachers came our way in another category, as parent or LEA governors, say, or if one or more governors in one of these categories happened to work in the school as support staff, we should probably decide not to co-opt more. Indeed, I have quite a lot of letters from schools where the balance has been upset in this way. In these schools it has come to be seen as a problem that too many governors are employed in the school, not just because of the dependence factor but because in practical terms there are too many ineligible or unavailable for various tasks. But from the rest of your letter I gather that you don't have this problem, so it is just something to keep an eye on, bearing in mind especially that next school year you will have an elected non-teaching-staff governor as well as having your co-opted school keeper - unless he resigns - for his remaining three years of office.