I don't know how that stand is constructed, but it seems like a very poor design to pad the seat with material that would soak up water. I'd take it off and replace it with a simple thick layer of closed cell foam - the dense stuff that is soft enough to pad, but won't pack completely when you sit on it. For example, I have a couple of old sleeping pads - the thin closed cell type. I'd use one or two layers of that stuff glued or stapled on. The outdoor shops also sell sitting pads made of this stuff.

The closed cell foam will get wet on the surface but won't actually soak up water. If you just wipe it off, you're good to go.

I have one climbing stand that I use that doesn't have a pad on the seat. I just carry a simple piece of the above described foam and just sit on it.