If you have dual propane tanks, the tanks alone will add 60lbs full. A single battery is usually right around 50lbs. Depending on your trailer, you have a tanks that add tongue weight as well (almost all do) So if you are carrying any water (waste/gray/fresh) you are likely adding tongue weight. The problem is that usually there is more stuff added to trailers in front of the axle rather than behind. Most have pass through storage up front, H20 heater, and fresh water. Go out with those with any amount of stuff, and tongue weight suffers.
The LR3 has no problem being a class III hitch, but the problem comes in that class three has a tongue weight rating of 500lbs. It is not a Class IV, which has a tongue rating of up to 1k lbs. (Althought some can be slightly more.) The stamping on the single part of 770lbs doesn't necessarily make the system (truck and all) capable of carrying that much weight safely.
Sway is almost always an unsafe loading condition, or an unsafe trailer mechanical condition. You won't have sway if your trailer is loaded correctly, hitched correctly, and in good mechanical shape. If you overlaod any rating. . .sway is usually the result and warning sign.