Maintaining LM2596 output voltage with switch
Is this a good method at 'stabilizing' the load voltage when using a
  regulator and a switch?


All your problems come from the extra LC you have added namely L2 and C4+C5. Together they convert a low pass filter into hyper resonant tuned circuit that resonates at 3.85 kHz and has a Q factor of nearly 70. Any step change in the load will cause significant ringing and over-voltage problems on your output.

It seems you are going down the road of setting a spider to catch a fly then sending in the bird to catch the spider. You need to step back and work out why you have introduced L2.

BTW - don't try using this within the feedback loop of the switching chip because you'll just make an oscillator.