In practice, the holes do not diminish the quality of bowling. But you can't prove that until the shared-use arrangement is already under way. If your priority target is a putting green, you will point out the dozens, if not hundreds, of public and private places in the immediate area are reserved for golf, with not a single site this side of Timbuctoo set aside for the serious sport of 6-Wicket Tournament Croquet. You will further note that 6-Wicket Tournament Croquet requires less than one tenth the territory per player that golf requires, on average. A USCA video would be excellent for this purpose.
Nobody likes to be pressured.
Just make the point. You will try to find out what the Key Bureaucrat wants and needs, and try to accommodate it. Your croquet group could provide the Park Department with a solution to this problem. The machinery of the public bureaucracy is never proactive, always reactive. If one avenue dead ends, go down another.