Hey, Iris.
So, pro dye wont work any differently in regards to stiffness, but you will see a noticeable improvement in evenness of coverage, depending on how you're applying it. I think that if you're making the leather stiff, you're applying it too heavily. This is a very common occurrence with dip dyeing and heavy application with a dauber. Its better to sneak up on your color with a few lighter applications than one heavy one. Easier to control and more even results. BTW, blue is one of the most difficult colors to use, apart form purple which is the worst.
As for oil, there is a great debate on whether or not its needed. Personally, I don't think it is. Maybe for saddles and horse tack, but for most things its simply not needed. People slap that crap all over their projects and then wonder why it looks dark or splotchy. However, I DO oil lightly between coats of dye when I use an airbrush because it helps set the previous color and helps the new layer absorb better. Buddy of mine gave me that tip and it works great on fades. Airbrush can be very topical and anything to make it sink in more helps.
The type of alcohol you use to thin your dye can affect stiffness, so I recommend denatured alcohol. That's what is in the dyes, and its the best for the pigment as well. Other kinds will work but your mileage may vary. Lastly, the type and thickness of the leather makes a difference as well. Thin leather will get crispy pretty easily while thicker leather maintains its suppleness longer. The alcohol in the dye affects the fillers in the leather, fats, collagen etc., so the thinner the leather, the lighter the coat should be.
If you can't avoid saturating the leather, replacing some of the suppleness with oil will help, but you must account for the change in color when you dye or it will be darker than you intended during dyeing. My suggestion is to look at the method you use to apply it and experiment with others until you find what works best for you.