Lye, Sodium or potassium hydroxide is extremely basic, pH is way up there in the 12-14 range. Baking Soda - Arm &Hammer stuff sold in the supermarket, is about pH neutral, around 7' Pure nitric or sulfuric is way low - acidic - around pH 1-2 range. I have not tested LMF Barrel browner/degreaser, but it is probably in the 4-5 range.
So you want to avoid extremely high or low pH if possible. And to neutralize the acid of LMF browner, use either household ammonia, pH 8-9, or baking soda, pH neutral.
If it were I, and having lost all the skin on both hands at one time, once with sulfuric, and once with nitric (slow learner!), I would first apply some household ammonia, even the "sudsy" kind, then dissolve some baking soda in warm water, and scrub thoroughly with it. That SHOULD neutralize the LMF.
Good luck with the process - it does make a lovely barrel in the end.