I would bake the clay to harden it faster, just as a clay pot or cup, and break it up and grind it into a powder to mix in the soil (cant really grind up clay in putty form). I just don't know how long to bake it though. As far as soil, what type?
I don't think you really want to harden/fire the clay. I believe the oven use is just a shortcut to drive out the nitrogen compounds (and maybe a few others). These would eventually get converted to some sort of volatile via microbial action in the standard flood/dessicate method.
Also, I thought typically the clay(and other additives) were added after mineralizing the soil, but before putting in the tank?
If you just want to get a powder form to add to the soil, let it dry out, and then smash/grind it up. smush it out into thin sheets if you need to. I've only bought clay online once in the past, but it came powdered (back in the day when laterite was the trendy thing), and I think powdered is the preferred way to store/ship clays (pointless to pay extra shipping for water). If not, just break it into clumps and smash it into thin sheets, should dry out quick enough, then just bust them up and get an easy to mix powder. If that's too much work, you could consider forcing it through a screen/sieve, to make granules, let them dry, and then just mix them in. won't be quite as uniform as mixing in a powder, but I don't think it will matter much.