You could try the surface washing, but dunking it even with a liner is not really as scary as it sounds. Well, before I say that, what color is the lining? If it's dark, you may have more issue with the lining. But as long as you dunk it, wash with a gentle cloth, then dry it inside and out/shape it, I think you should be ok. You would dunk it then stuff it with old towels so it fills out it's shape, and keep an eye on it. Some of my drying projects took more than one rotation of stuffing.
The ONLY thing I hesitate on is the fact that Black Rock is mostly carnuba wax. I'm not sure what that would do to the leather. Past history with almost pure carnuba wax used on my car's washcloths meant some sticky washing later. I don't think it would hurt the bag, I just wonder if it would draw the wax out of the leather pores and just smear it around.
One other option: keep doing the BlackRock as directed on the label, concentrating on that spot. The BlackRock is SUPPOSED to clean it, and more than one iteration of proper cleaning may draw out whatever is darkening the spot.