Paying for the partially composted materials why the heck would you screen any of it out? Smaller than a 2x4 leave it alone LOL!
MGOCPM (potting mix) contains soil and a large portion of organic material (55-65% by volume). Sphagnum peat moss, composted bark fines, leaves, twigs, wood chips etc. and "pasteurized poultry litter" (cooked chicken crap).
I have repeatedly tanked this material straight from the bag to the tank without earth shattering, life ending drama. Algae has yet to ruin a tank here either. Controlling algae is as simple as controlling the light energy supplied and planting the tank within a day or two of flooding it. I do like adding frogbit starting out for the dual benefit of light control and the way this plant removes any leaching nutrients from the water column early on.
Straight from bag to tank I've done a dozen systems this way so I don't see reason for concern over the large ratio organic volume.