I'd be getting a couple of other quotes at the very least. Possible the C of E have 'Approved Contractors' who they use for their work, and I would hazard a guess, any invoice submitted will be nodded through for payment, regardless of the amount, which may lead to prices being somewhat inflated, as there is no competition to get the work.
Alternatively, tell them you are happy with your half of the chimney, and will not be authorising any work to be done on your half. Also put them on notice that any damage, or works to your half necessitated by anyone working on behalf of the C of E on their side will be deemed their responsibility, and will need to be made good at their expense.