I can't see how a single-axle trailer would ever be more stable than a dual-axle, and EVERY towed trailer has most of its weight on the tires. Perhaps you should take possession of the trailer and see what it is like when camped--might save some money to get other goodies for camping.
As to the king-pin stabilizer, not a lot of fivers use them that I notice. They are very heavy and awkward to carry around. I had one and saw no difference using it or not, but others have said they work. Tire chocks are a must--one of the best brands to use is the Roto-Chok I hear. I use BAL chocks on both sides and that is the only thing I use to prevent motion--they work well for us.
It is a personal choice, and everyone is different as to what they feel in trailer movement.
Good luck and have a great time with the new trailer.
'05 36TK3 Mobile Suites
'10 F450 new to me
'09 F450 Lariat 4x4--died