I felt my original answer got messy. The simplest answer is that you already had it. I experimented with variations of your 12T/24T build and experienced varying degrees of the stiffness you mentioned. It became evident that the location of the gears was affecting how much the side pieces were pinching the gears even though the actual gear mesh remained well within tolerances. Stabilizing the brackets, thus the distance between them, eliminated any and all stiffness.
I also threw together a few more 1:2 ratio gear configurations for reference:
Inspired by the design from Yoshihito Isogawa posted by Uli, I couldn't resist tweaking it in an attempt to make it stronger and easier to build off of.
Inspired by the answer that Eilon gave regarding Technic chain, I evidently proved myself wrong and came up with this.