You have a completed a quick-and-dirty analysis that agrees with your sponsor on all but one key point. Your sponsor wants you act now - to design and build a specific solution identified by that sponsor's boss. The data in that one key point shows that the preferred solution will not be accepted by those who must implement it. You have two choices:

(a) get agreement for more analysis work before proceeding, or

(b) proceed and build in additional analysis along the way hoping that feedback will change either the executive sponsor's position or the implementers' position. Choose a position and explain how you will move forward.