My guestimate (I have not tried your method) would be the barrel has already flash rusted by the time you have applied the rust inhibitor. Now being you are using a woodburner, most likely the surrounding air is dry & almost no humidity in the air at all, so this may be very minor rusting.
I do know for a fact that if you clean a barrel with hot water & let it dry, it imediately starts to rust once the water is gone & the longer it sits prior to a rust inhibitor being applied, the more it rusts.
Now we are talking micropic forms of rust starting and a good rust inhibitor will kill this imediately, thus no harm is done.
I did however clean one this way back in the mid 70's, bucket & barrel in hot soapy water cleaning & flushing it in & out, took the barrel out & rinsed it with hot water, left it to dry. Well, I got distracted somehow before I got finished with it (don't remember what happened) & never got back to it til the next day before realizing I had not protected the bore after rinsing it. That bore to this day is still etched. Now it didn't rust bad mind you, but it doesn't have that mirror bore look & never will have again. And it didn't affect the accuracy of it at all, well, best I could tell off a bench.
But for me, No, I would not heat the barrel. Nor do I use hot water in one, as cold water & some Ballistol afterwards does fine for me. Lots of guys use hot water & like that method & that is fine.....
Lots of dif. ways of cleaning them, ya have to find the one You like & waht works for You....
:thumbsup: