Hurst shifter box too much play?

The side-slop, if it is not affecting the 2-3 shift, I wouldn't take the time to change it, unless the rattle becomes unbearable. That's what the wavy plate is for; to prevent rattle.That plate is pretty hard for how thin it is. My shifter is from the 70's, and that plate is not worn to any meaningful degree. Yeah the last time I looked,it had shiny spots, but not meaningful wear. However the waves do tend to flatten out over 40plus years . And although it's somewhat difficult to reshape the plate,it can be done.
As to the actual stick-play, "ssba" hit that pretty good.
If you decide to take the mech apart,take pictures, lots of pics.For such a simple device, it's pretty complicated to reassemble correctly. It's easy to put one shim in wrong, or one lever in,out-of-place, and then you get to start over. Make sure to bench-shift it several times to make sure you got it right!