You done good in degreasing the barrel.
Birchwood Casey Plum Brown doesn't work well if there is any grease (or fingerprints) on the steel.
When you used steel wool, unless you totally degreased/dewaxed it, it probably has put a layer of wax on the metal so you should resand and degrease the barrel before trying to use the browning agent again.
OK. Now down to what to do.
Your temperature was about right. The fluid is supposed to bubble and boil off like it did.
If the temperature is correct, the solution will immediately form a brown coating.
If the temperature is too hot the solution will spatter forming a spotty black/brown crust that doesn't coat evenly. If this happens, continue to apply the solution until it forms the (fairly) even color.
If the temperature is too cool, it will not brown the surface and it will probably lay down a layer of copper plating. If this happens the only way to fix it is to sand off the copper.
After you have browned the part, let it cool. Then use water and some clean paper towels to wash off the hardened deposits.
Plum Brown doesn't build up enough rust to need "carding" (hard rubbing with a course cloth or very light rubbing with fine steel wool).
When the deposits are washed off with the water and towels, reheat the barrel and apply another coat of the Plum Brown just as you did before.
After applying each coat use water to wash off the crusty deposits.
I've used this product on several barrels and I've found that applying less than 3 full coats always results in a streaky appearance.
That said, plan on applying at least 4 coats.
After the final coating and washing, dry the barrel and coat it with a layer of oil.
Almost any kind of oil will work but regular motor oil seems to work much better than a vegetable based oil.
After coating the barrel with oil, set the wet, oily barrel aside and let it age for at least 24 hours.
This aging will darken the color of the coating a lot. It will also cause the streaks to blend in with one another to give a much nicer look than you thought you were going to have to live with.
As I said, I've used BC Plum Brown for years but I now use it only for small parts like locks, hammers, cocks, steel butt plates etc.
On those small parts it works very nicely.