It shouldn't make any difference. They both make up alkaline solutions when dissolved in water. It's the pH that neutralizes the browning solution. The instructions that came with my bottle of LM degreaser and browner says to use baking soda. Dilute lye should do the same thing. I washed my rifle with baking soda and it turned out just fine.
I don't want to hijak this thread, but I thought some might be interested in my experiences with LM Degreaser and Browning solution.
The first coat turned the barrel an ugly mottled color. I let the barrel set in my bathroom with a hot steam humidifier running on the high setting. The humidifier made it feel like a sauna. I'm not sure, but this may have produced a darker brown, which I prefer over a lighter chocholate brown. The next coat wasn't much better. The third coat started turning it a lighter brown color, but there was still some uneveness in the color. The Fourth coat started evening out the color and it started turning it darker. The fifth and last coat turned it darker yet, and evened out the color to my satifaction. I have posted a picture of the results a few threads down. I would use it again. It was pretty easy to use and I'm pleased with the results.