Baking soda is a acid neutralizer. So is water, if its pure enough. A lot of water comes out of wells, and has high mineral content. That can form acids. That is why they want you to add baking soda to the water. Then the water neutralizes and rinses away the acid in the browning solution. If you put too much baking soda in the water, it just helps to neutralize the acids faster. No harm done. A couple of tablespoon to a bucket of water should be enough to do the job on your barrel.
I simply used tap water and a vigorous scrubbing with a rag to get the residue of my browning solution off the barrel when I was done, using Birchwood Casey Plum Brown, which is different from the LMF. But, both are acids( they have to be!) so you have the same cleaning issue with both products.
When you dry the barrel, oil the barrel and tang and any other parts you browned with the solution to the point of excess oil dripping off the parts. Put some paper down to catch the oil, and let them sit overnight. The oil will help Set( stabilize) the brown finish.