Before I post a response to this question, I have to say that I have never fire lapped a barrel, nor have I hand lapped one. This is my opinion that is based on reading material, and listening to people who have done it.
I considered fire lapping a Winchester Model 70 Featherweight in 300WSM that was copper fouling terribly. I talked to several gunsmiths that advised me NOT to do it. When you fire lap a barrel, you run a chance of moving the throat forward. This would only serve to compound an existing problem with a Handi Rifle.
I have also been advised by custom barrel makers to not use abrasive compounds on a patch, pushed through the barrel with a rod. The reason is this; if you lap a barrel in this way, you 'round off' the edges of your rifling. This can affect the way spin is imparted to the bullet as it travels down the barrel.
Now, all this being said, I do know that there are people on this board who have run a patch down their bore with Flitz metal polish on it, and claim that it helped to 'break in' their rifle. I don't doubt it either. I think Flitz is on the extrememly fine end of polishers.
Good luck and let us know what you end up doing.