I can't believe you backwashed 10 minutes and still got dirty looking water - sure you weren't looking at it through a microscope?????
I backwash 3 minutes at worst (like after killing an algae outbreak and cleaning that up) before seeing clean water and that's on a way bigger pool.
if you added a small amount of floculant and bleach to your "collecting pool" and used a small cover type pump to recirculate the water for a bit (to mix floculant), then all crap should settle if left standing for a day or two and no algae would take hold due to bleach, then you can set in a pump - raised off the now dirty floor several inches, and pump the clean water back to the pool, then lower the pump to pump the sludge onto those brown spots on your lawn