Eric,
I have used LMF for a number of rifles and have never had your issue. However, I do it differently than you.
First, I clean the barrel VERY WELL. First hot soapy water, then lacquer thinner, then another bath with hot soapy water, all while wearing canvas gloves. I don't believe the assertion that the LMF is a "degreaser".
Second, I wipe each flat with a single swipe of a cotton patch soaked in LMF so that the surface is merely dampened. I do not allow droplets of solution to appear on the barrel flat.
Third, I put the barrel in a damp box I made that is equipped with a pair of incandescent light bulbs and a pan of water with a cotton rag in it as a wick. I let the agent work for a full 12 hours.
Fourth, I card the surface with a piece of 100% wool blanket.
These steps are repeated every 12 hours for 4 to 5 days. The first couple of cardings don't seem to have much in the way of rust formed on the surface. Don't worry, be happy. Subsequent cardings exhibit more rust but much of it seems to wipe off. Don't worry, be happy. By the end of the week you will have a matte textured barrel with a deep even brown.
I then wash the barrel thoroughly with hot soapy water, lot's of it! I have never used a neutralizing agent, just lot's of water to dilute and wash away any remaining browning agent. I dry the barrel, using WD-40 as a final drying agent. Then I use a propane torch to heat the barrel until bee's wax will melt on contact and I melt bee's wax over the entire barrel surface while heating. Eventually the barrel is dripping with liquid wax all over. I take a rag and wipe off the excess wax.
Barrel done.
That's how I do it. If you like the look I get then feel free to use the same process.
Best Regards,
JMC