I was one of the first to do the conversion (take a look at
this thread, and especially post #5), and I, too, was worried about shaking the trailer. However, the shaking would come from rock-hard tires (the bigger tires can run at 65 psi, rather than only 50 psi for the smaller ones), so my answer was to not to run them at 65 psi. Even at 50 psi, the bigger tires have considerably more carrying capacity than the 14 inch tires. I run mine at 60 psi, but that is mostly for Intersate driving. I would probably soften them a bit if I was about to journey over a lot of bad (meaning bumpy) road.
By the way, "upgrading" the axle (meaning a change to a stiffer axle) would only make the shaking worse, so I don't understand the reasoning behind this suggestion.
Bill