Doesn't look like you have room, but there's a way to make a drawer slide machine a lot more rigid. Use two slides on each side, at opposite ends of the travel, opening towards the center. That way, when one slide is fully extended, the other is full retracted. Makes it a lot more rigid.
Downside is that it takes more space, more slides, and installation is rather tricky, as you probably need to use bolts and nuts to mount the slides.