I also use the IKEA KALLAX shelving units (see my system photos).
My solution was quit simple: I added a piece of 1/4" ply to the back of the shelf - cut to fit, glued and screwed. The resulting units were DRAMATICALLY STIFFER than the basic unit.
I now have simple, solid and very affordable storage for ~ 2400 LPs.
One more 'tweak': I added 2-lengths of inexpensive, black pipe-insulation to the back of each compartment. This insulation keeps the LPs forward in the box, at just the right place ;-). It is sold in various diameters, so test to see what you prefer...
I hope this helps!