You'd be safer with a "foam type" leather cleaner than Fine Fabric Shampoo (one of my favorite products, to be clear). I would only vacuum or towel extract afterward.
I know its going to take longer to do it that way, and the product costs more. But you'll be doing what is in your customer's best interests.
I used my pmf internal jet tool on a couple. Be careful to not over wet the skin and make multiple dry strokes.
In fact I use that tool as my own personal "FlowBee" for my own hair when I'm on the run.
Anything safe for wool and a hand tool would be fine as long as the hide is in good shape. If the hide is dry or torn, or hair pulls out, pass it on to someone with experience.
These sell for $175 or less, but I'd still charge $4.50 PSF to clean one.