good question and I too know a little more thanks for asking
I have not done crazy quilting pith but I have done it freehand before. I found you needed some kind of light weight base fabric because you do not place things on the straight of grain when crazy quilting and you need that base to keep things straight and flat. You could use a light weight tearaway under it when doing the designs so the stich out better. It can be floated under the square after the piecing is done and before the designs that will need it..
When I have done these, I have just used a scrap of fabric as a base and skipped the stabilizer altogether. This is for the piece in the hoop (PITH) and the crazy quilt patterns that do not have extensive stitching. Some of these types of designs have additional small designs stitched on them, I think they would need a stabilizer.
When I've done ITH crazy quilt blocks (where you piece and embellish with one hooping), I've used 2 layers of water soluble (the paper looking stuff, not the stuff that looks like plastic wrap.) You certainly don't want to make the mistake of using tear away - hmmmm, does that sound like experience talking?
LOL *'s for all.
I assume you are going to use a base fabric. If so I would starch your base real heavy and that way you could use a wash away and then not have any pokies sticking out or have anything to remove after you add your pretty stitches.