More of a question than an answer, but should spoke tension be checked as a possible factor? Compare tension in the flexy wheel to tension in the other wheel (Williams) that the OP has?
(I believe I could output 1000 w for about 1/1000th of a second, so no answers based on personal experience here.)
Spoke tension does not affect wheel stiffness assuming a minimally taut spoke at least.
Spoke count and as FB stated, spoke cross sectional area are the main determinants along with bracing angles of the spokes. The higher the angle from the vertical, the stiffer the wheel. And it is a squared function, so fewer spokes on the high angle(NDS) side make a big difference. Are there very few spokes in this wheel. Are there only 1/2 the number of spokes on the NDS as on the DS? If either is true, that could easily be the answer.
Also a very stiff rim will rub the brake more readily than a flexible rim. That is because the flexible rim bends across its mid-line and the upper half of the wheel doesn't move very much. A stiff rim doesn't bend so lateral force at the road or mid line gets transmitted all the way to the top. With a very stiff rim, you need very stiff spoking to prevent the wheel from tilting around its center diameter axis.