Look at your BOAT registration card. Nothing on it identifies the outboard that is on it, and nothing on the outboard registration slip says that you must carry it at all times.
If you wanted to borrow one of Babler's 20hp Yammie's for a few days then you could do so without going up to the DMV and paying an assortment of fees.
And any MOWP officer that writes you up for it is just being a thug, plain and simple, unless they are investigating a theft......which they NEVER do, EVER.