[Martin Smith]

There you go, some kind of triangulation at the base of the posts would be essential and would reduce most of the flex to a negligible amount. The only remaining concern would be the rigidity of the top cross bar, which could twist around its two mounting points. Hard to say if that would end up being a problem or not. Uncharted waters here I think. Perhaps have a plan B if you build it, in which you add one more cam lock on each side so the top can't rotate (leading to the wheel moving up and down).

C-channel can be 'heavy duty' but that doesn't mean it's rigid. Consider that a much beefier version of what you're showing is used as a twist-beam for the rear suspensions on many cars!