Keeping the dirt in that Rubbermaid container moist as a wrung out sponge, with the lid on in a warm spot would decompose the roots fairly quickly. Over moistening with no drainage however will result in dirt soup that may or may not smell. Coffee grounds & few handfuls of crushed dry leaves mixed thoroughly would be good additives, again provided over moistening does not occur.
Hope that helps!

Great thanks for the advice paleogardener :) How long do you think it would take for the roots to decompose? Just a rough estimate and then I will do a test on my own.

Also... are composting rates slower if you have more material in the bin verses a smaller amount? I am asking this so that I know the test size I need to run for it.

Thank you!

The roots should be so fine (because microgreens) that it shouldn't take too long to become either composted or fragile/fragmented enough to not matter.

One of the best things you can do is get your hands in there and break up the soil after you grow the micros in order to break up the mat of roots. The roots should be quite fine, but there should be a huge amount of them which may aggregate the growing medium into chunks/sheets. You don't have to break it up into a fine textured medium, but you want to get some of the larger chunks broken up.

you can always get a cheap food-processor, or blender at a garage sale, or the Salvation army.
they always have those things cheap.
the more surface area, the more air hits it, the faster it should break down ( i would think)

Great thank you both so much for the advice! I am going to test this compost out and see what happens. I will post the results that I find so that others can use the final information as well :)

How long it takes to decompose those roots is dependent on several variables sun exposure, aeration, right moisture etc. Drying the roots in the sun for a day or freezing them will speed the process. So for time I would expect no identifiable roots inside of a month, provided some attention is given to the process.