The longer you wait the harder it gets. If you wait until the project is completed, its too late. Do it now regardless of the embarrassment. The project purpose statement you wrote and understand will be seen differently by everyone else. If you do something that you are sure will meet everyone's approval, someone will not like it.
Measurable benefits are real.
Intangible benefits are not measurable, thus intangible benefits are not real. Intangible benefits are real if you can prove that they are real. Anyone who can work effectively on a project part-time certainly does not have enough to do now. If a boss will not give a worker a full-time job, you shouldn't either. If the project participant has a time conflict, the work given by the full-time boss will not suffer.