I'm willing to bet you could gather some more customer feedback, but let's consider the item you mentioned:
"I don't offer "unlimited" everything"
My guess of what the customer is telling you is that because they don't understand the technical side of things, they are worried they will end up paying more in fees / overages. Same reason cell phone companies are offering unlimited plans - they provide comfort that you won't go over.
How do hostgator and many others offer Unlimited Disk Storage & Bandwith for such a low price? Because they know that even if they posted real limits, most customers won't even go close to those, and I bet they deal directly with those that do to optimize. Saying "Unlimited" is worth the (likely small) risk to close the deal. Listen to the customer, they're telling you what they want!
Something to consider, not sure how they would respond - what if you charged by the size of the organization (churches : church members)? That would loosely approximate bandwidth, and probably make sense to your customer.
One other thing I would point out, a little customization goes a long way. Can you offer to customize the site for their organization, or their specific needs, for low cost or free? You want a recurring customer, right? Put in a little work up front (show them their possible site?) and you may have a better chance of closing the deal. IMO, for non-technical clients, it's much easier to sell them on something they can see.
I wouldn't give up on this just yet, I think it's a nice niche, but you really need to understand the customer needs and why they would or wouldn't make a purchase decision.