I hope you either have a really light camper or a really heavy duty truck. Load the camper with your normal travel load, full tanks, and whomever travels with you, then weigh the front axle. Measure the horizontal distance from the front axle to the center of the rail or baskets of the carrier. Look up the wheelbase of your truck. Divide the front axle-to-carrier distance by the wheelbase. Multiply that quotient by the front axle weight. Deduct that product from the Gross Axle Weight Rating listed on the sticker on the door jamb. The difference is the weight of motorcycle AND carrier you can safely carry.
Gross Axle Weight Rating - (Front Axle-To-Carrier Distance / Wheelbase) Axle Weight = Weight of Motorcycle AND Carrier
or
G - (R / W)A = M
where
G = Gross Axle Weight Rating
R = Front Axle-To-Carrier Distance
W = Wheelbase
A = Axle Weight
M = Weight of Motorcycle AND Carrier
My guess is that unless you have a really heavy duty truck and/or a really tail-heavy camper, you won't have the carrying capacity on the front axle to safely load 350 pounds of motorcycle and carrier. If you are close, you can try redistributing the load in the camper to free up some front axle capacity. Alternatively, you could seek out a lighter weight motorcycle, say, a CT70 would by cool, or one of the Chinese clones.
EDIT: Front-mount hitch receivers for pick-ups are fairly common. There are a host of accessoroes that mount in recievers, winches, cooking grill, tables, bicycle carriers, cargo carriers, folding bench seats with rod holders for surf fishing, cargo platforms, tent poles, radio antennas and towers, tall poles for lanterns with polished stainless reflectors to attract the bugs away from the outdoor kitchen while still lighting the kitchen area, poles to hold solar showers and enclosures, etc.. Lots of folks who tow boats or travel trailers find them handy for "backing" trailers down curved ramps or into tricky campsites.
