I just use the brush for soaping and scrubbing, and the hose with the standard spray nozzle for wetting and rinsing. The bristles are stiff enough to scrub the cloth, but not so stiff that they end up grinding the dirt into the Tipi cover fabric.
Anyway, you've got your tools and your bright sunny day. It's time to roll up your sleeves and get to work. You can lay them out on the lawn and scrub it there, rinse it off, and then hang it vertically to dry. Since I had about fifteen feet of garage wall handy, I nailed up some scrap conduit so I could leave the sides hanging for as long as it took to dry.
If you haven't such a handy expanse of wall, or a really strong clothesline, you can of course set the tipi up in the usual way. But try to keep the sides (including the sod flaps) away from contact with the ground until they dry. Scrub the cloth gently. I use a side-to-side motion of the brush to maximize the number of bristles in action at each pass. It seems to do a better job than moving the bristles up and down.
Remember that using excessive pressure or a stiffer brush can result in the dirt being ground deeper into the cloth, rather than being floated away from it.