My relentless pursuit to cure trailer sway - the dancing TT

Have you had the trailer tires balanced? Have you checked them for tread separation?

I'd say forget the scales...(I know this is controversial here)

Take this trailer to a PROFESSIONAL heavy duty Trailer alignment place that works on big rigs and have the axles aligned and the Wheels checked for Roundness and balance and the Tires Checked for tread separation.

My guess is therein lies your bogey man.

With your truck that trailer should pull and track straight as an arrow.