I suggest setting the milestones and deliverables in the response to the customer so that they are well within your resource constraints. That way the customer will have clear expectations on what you will deliver and when and you can plan your resources accordingly.
Our mentor advised us that anyone working in a large corporate that would reject suggestions of managing personal risks is unlikely a good long-term champion to work with. So from the point of risk management for the individual champion on the other side, plan for a longer term, higher-volume project, but start with smaller volume with greater rate of success will get more buy-in from others in corporate. (you know it will work, we know it will work, but to help you get more support, let's pace it so make sure others will be on board too)
Create a gantt style project plan with gradual ramp and time set aside to define and agree on acceptance criteria as well as time set aside for formal sign off that success has been reached. Break it into manageable chunks where you can show a “win” every week. The customer also wants the project to succeed so they should be on board with this. We have lots of Fortune 20 and 50 customers - the key is just take it slow and document everything so your champion can clearly articulate progress and goals to their peers and managers. Good luck!