I would speak to the C of E property dept and ask what the **** it's all about. May well be the builders trying it on or perhaps the work is to be done and they were simply meant to inform you they were carrying out work to their side of the stack and sent a quote by mistake.
You could tell them it is a disgrace for the church to pray on people like this!
Regarding the payments to the local parish mentioned above I am pretty sure it relates to work on the local parish church not their rentals.
apart from anything else the quote seems high to me, obviously it varies area to area but still 5k for 11 courses and a couple of pots?
is it really difficult to get to?
personally id take woodys advice, write them a reply
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.