I wouldn't be so quick to blame the weight of the truck. While weight does play a factor when coping with sway in a swaying event, I've never heard of the lack of TV weight causing that sway. Really that's your problem, trailer sway. Trailer sway shouldn't exist ever regardless of TV. There's something within the setup of the combo that's causing or allowing sway to happen. Commonly it's the hitch setup, tongue weight, or lack of sway control.
A smaller trailer really could just be a band aid that masks the true problem. There's a chance that the problem can rear its ugly head in the future if you don't truly understand why your last setup swayed. For example, if the problem was too light of a tongue weight and you load the new trailer in a similar fashion, the you'll have the sway problems once again. This time with a trailer that well within your limits.
Back to the TV weight, a heavier truck still may have not been the answer to the last combo. Due to the nature of sway, even the biggest trucks can be flipped by relative small amounts of weights. Furthermore, there are tons of vehicles out there smaller than you truck, pulling trailers trailers many times their weight without any sway. So the weight of the TV isn't the end all be all. Sway prevention is really what's key.
