Without having had a closer look, and with only limited experience, but I would say you do not need to filter that clay. If it comes from deep in the ground, it will not contain no debris of considerable size. Like roots or the like. Single pieces and larger rocks will also come out while wedging. Filtering is lots of work. Unless I want a super smooth surface, I dont.
I've also seen people filter the dry, crushed clay, but do this outside. Clay is easy to crush once it has frozen thoroughly and thawed again, then dried.
If you have problems with hard clay, simply soak it for a longer time. 7 to 10 days will make for soft clay usually even if it was dried completely.
for tempura, I have tried sand, crushed pottery and horse manure. The manure gives best workability and will result in porous, light, and heat tolerant pots. The sand makes for dense, heavy pottery, try out how much is good. Crushed pottery is probably ideal for many purposes, but it is so much work to crush and sieve the stuff.
I also got away with non-tempered clay more often than not.
Do dehydrate my clay, I fill it in a cotton bag (got it in hte army for the laundry) and hang it in a tree, the water will drip out within a couple of days. Professional potters seem too use plaster plates, thick, dense fabric will do as well.
Also, I would recommend processing quite a bit of clay at a time, as the process is messy and the product stores well
