At home the hitch and bars are in the garage. When camping the hitch stays locked in the receiver and the bars go on the A frame. (yes the frame up by the batteries and propane tanks) If I feel unsure about the campground I put the bars in the front pass through storage.
In your case since you don't have a garage, yes you can store everything in the front storage of the trailer. But I imagine it will be a chore getting that heavy greasy thing in there.
I would probably get a wheeled under bed tote
Link I use one of these to hold random junk in my storage.
In use you can lift up the end of the tote enough to clear the bottom of the door frame and pull it out from the compartment about a foot. Place the hitch head in the tote as close to the door as you can. Now lift up the end again and slide the tote back into the storage compartment. The tote will slide in easily because the rear wheels (furthest away from the door) will roll on the floor.
The wheels closest to the door will be useless as the hitch will flex the plastic bottom and sit on the floor. But the hitch and all it's grease will be contained in the tote and not on your other storage items.
