I use exotic leathers in my holster making. I've attached a few pics as examples.
You'll find that if you daub or sponge your dye on, the exotics (such as beavertail) will soak up too much dye, and it will literally take a good week and a half or two for it to dry completely.
For those leathers, I tend to use an airbrush, and apply the dye VERY sparingly, keeping the airbrush on a wide pattern, about 2 or 3 inches off the work, so that the dye doesn't evaporate its carrier solution before the solids have a chance to get into the leather.
This way, your exotic stays dry, and you can laminate it to a heavier backing leather (7 - 8 ounce Veg Tanned, or Horsehide) and finish constructing your holster. My own preference for laminating is using a spray contact adhesive (3M makes a good product) because you have basically no cure time with contact cement, but you can also use Gorilla Glue, but do so VERY SPARINGLY, as it expands as it cures. :-)
Feel free to drop a PM if you have questions.