My experience is that leather is sort of like Pampers material. By that I mean if you dye the finished side in a reasonable manner (not dunking, soaking), then the dye stays on the finished side.
On the other hand, if you put something (dye, oil, water) on the suede side, it transfers fast to the finished side like a sponge.
I have read on forums where folks will say, "I am thinking about putting a turbo on my car. What do I need to do to get there?" The best response is almost always, "Sell your car and buy the turbo version." In this case, if it really bothers you, sell that bag and buy one that suits you. I think you'll be happier in the long run. I only see unhappiness in trying to dye your existing bag.
I am no expert, but I hope this helps. God Bless, Marc