Some campgrounds in bear country won't allow grills to be left outside.
I would guess most RVers store their grills in the basement of their RV and there are no problems with the bear smelling it and trying to get in. Same as in vehicles, although in some national parks, particularly Yosemite, if a bear even 'sees' the grill inside the car it has happened that it broke into the vehicle.
Perhaps after storing the grill in your Rubbermaid, spray the outside of the container with some offensive stuff like ant spray or OFF. Perhaps this would deter them. Or...just store it and hope for the best. :) Good luck!