I also have no experience but a couple ideas. If you're not looking for "production" output, you might try a hybrid hand-painting method. Print out the design on a film, stylus (yes, that's a verb now) it into cased leather, then dye the design in with a brush (or more likely several brushes of varying sizes). Some DIYers do this with t-shirts and textile ink rather than just building a frame and buying photo-resist.
I would also probably try proper silk-screening. As Sylvia said, the inks used for screen printing are really more like a paste. Leather dye might need some sort of thickener to keep it from flowing under the screen, feathering and/or smearing the image. I have essentially no experience at all with leather dying (just tried dying and burnishing my first edge last night -- only to figure out it was chrome-tanned leather!), so I don't know if a thickened dye would still be able to penetrate the surface and such, but I don't see the harm in experimentation!