Unsure if your still working on Cost per Contact. I had the same thoughts as you working within a public sector contact centre; we were not an cost efficient centre at the time, and while things have changed we needed to produce an incentivised cost recharge model to encourage investment from service area's to move into the contact centre.
Firstly, to get over the different costs via channels (mainly because the contact centre are also managing the authorities channel migration policy to cheaper channels) we calculate Cost per Work Minute rather than per contact. CpWM excludes time spent idling waiting for another contact.
We have a reporting system which allows us to classify each contact regardless of channel, breakdown total length, and associate it to a customer. We can therefore assign total work time by customer. Then to simplify what is actually quite a complex process we divide total time by customer, by total overheads.
Our calculator is used to project costs of prospective services, so will add a team leader in per x FTE advisors required, will anticipate staff costs dependant on skill required to complete the transaction, include an ICT overheads for software licences etc and also as our authority internally charge for space per desk, we include that as well.
Effectively our customers pay entirely for the service. The balance of cost is fluid and varies dependant on the demand, although we provide a projected cost to them for budgeting reasons. Our cost commitment to them is that we will, collaboratively manage their customer demand to a more cost effective channel over a 3-5 year plan. This means they can encourage customer contact while hopefully platforming total cost.
All of this is made possible with the use of an effective CpWM model. Hope this helps.