Keeping to scale is the problem.1/8" thick leather disks at 12:1 ends up being .010" thick.
The lightest weight of leather is 2oz. and is about .030". thick
I have used lizard skin to simulate rayskin on sheaths and Japanese handles.For a stacked handle perhaps frog, lizard, or snake skin would work. The thinnest grade of lamb skin might be OK for a 6:1 scale mini.
Or- you could tan up some mouse hides

- which I did with a white mouse for a mini wool covered viking sword sheath. I used Rep-tan from Tandy. It was OK, but did not really look like wool.
I experimented with making a stacked handle look from a solid material, but the only thing that had scale thickness was tan/brown micarta cut across the grain. The paper stuff did not show any striations, the linen was OK, and canvas too coarse. Model aircraft plywood looked sort of OK.
To make a stacked handle from thin pieces of leather, I would cut or punch out 1/4" pieces of your material. Then thread them on a sewing needle. Use a cork on each side to allow the stack to be squeezed in a light pressure clamp. Apply thin CA and repeat until it is fully permeated. Cut the corks off and chuck the needle in a dremel and sand into a round cylinder of the desired size and shape. Heat the needle to allow it to be pulled out of the handle stack. Re-drill the hole to match the tang on the mini.