Can I come watch when you backwash into a bucket?
DE is good for the soil, actually, it's great for the grass since it is what's used to kill grubs, so if you have grubs, you now have grub control.
Backwashing into mulch isn't going to go well, a gravel bed is much better.
Can you dig a 5" deep discharge bed for holding gravel, then have that spill over into the lawn or yard? that would be much better than washing all you mulch away every time you backwash.