... where visionaries, game changers, and challengers discuss business models
I would indeed proceed in a similar way:
In addition you might also want to think of a portfolio of business models. Which one is producing growth, which one is declining, which ones are emerging, etc.
I don't want to be to invasive, but what kind of business is this? There are a couple ways to decouple and re-integrate the different services. Are the services and organization geared toward products, region, business unit? Depending on how you group them could probably lead to more problems than solutions.
The business model canvas deals with two primary areas the customer proposition and the business operating model. If you draw a dotted line from the cost and revenue line through the value proposition you get the two components.
I would as suggested identify the key processes using the canvas as discussed by previous contributors. These need to core processes and should be the critical "level 1" processes. It is important that processes run horizontally across the canvas joining up the components of the canvas. Using this methodology with "swim-lanes" you can begin to identify common processes (or repeating process components) and services and authentic differences.
SOA Process Map Convention:
Business Process and SOA run horizontally across these two components "end to end" and "front to back" from the customer experience (known as the "line of visibility" or moments of truth) through the fulfillment to the line of automation that is the systems that support that process or service. The definition on this way of the core process of the Business model are at the core of SOA.
What this mapping does is to connect horizontally the customer segment through the value proposition to the key resources and activities including key partners. This done through a mapping convention based on "swim-lanes" happy to help with these if you wish.
In summary, the Common and Shared business process can be identified through the canvas as a horizontal end to end view.
A year on this may not be particularly helpful, but see what you think now.
SOA machinery is most powerful when it provides flexibility/agility. Working from this insight, what can you say about the value/urgency/need for flexibility in the various businesses -- the common needs would yield the most welcome results (and smooth the way over some of the subsequent rough spots).
I don't know your achitectural skill set, but business models are inspiration for engineering while not the actual engineering. Make sure you have a team with the architecture skills (turning good ideas into good blue prints -- in the systems and software senses). Often this means consultants if the company has not succeeded (or attempted) at major changes --- because the company engineers will be better at modifying their processes than they are at (potentially) throwing them away.
By now, you will have a good handle on the maturity assessment for the company and the businesses. You might want to contribute some lessons learned based on how the people have handled the changes so far.
You have an interesting situation. Best wishes.