I suppose you will have practiced forge welding in general. Make some good welds, and examine your mistakes.
You might also deliberately overheat a piece of steel so you know for sure what "burning" the part means. Lots of white sparks, unforgettable. Means the grains are surrounded with scale (iron oxide) and the whole piece will have no more strength than that scale. Overheating the billet, hence burning the steel, caused some of the problems with low-numbered 1903 Springfield receivers.
Forge welding modern steel is not so easy as forging wrought iron, which comes full of its own flux. I admit I am pretty low on expertise - made one forge weld in wrought iron, just before I burnt my first iron. Flux not needed.
Today some use 20-Mule Team Borax for steel, others one or the other proprietary flux. I've no idea which is best. But you must use some manner of high-temperature flux.
When you are finished with that barrel, each and every inch must be welded soundly.
That is mostly why they proof tested barrels in previous centuries, to test the quality of the weld.
Defects in the iron were much more common than today, but they did account for some failures.
I saw one illustration of someone "testing" his forge welded barrel by driving a tapered pin into a short cut-off length. That, I presume, determined if that short length was strong, but said absolutely nothing about the rest of the barrel.
You might want to read Harpers Ferry Armory and the New Technology; The Challenge of Change by Merritt Roe Smith, Cornell University Press copy- write 1977. This tells how musket barrels were actually made -no romantic stuff - and proof tested. Before they got a power hammer (helve hammer) loss in proof at Harpers Ferry sometimes hit 40%. By the use of a power hammer (and probably better iron) Springfield kept the loss down to about 12%. Maybe I'm in error about that 12%, I strongly suggest you read the book yourself. It is the only book I know that dispassionately covers forge welded barrels.
Apologies that I have mislaid a section of wrought iron barrel that burst at a defect while firing blanks at Friendship. Maxine Moss lent it to me, since she is long gone and they had Denise "retire" I don't know anyone there personally. Back to the barrel. It had quite a forging lap in it, which some fine fellow had brazed up. Well, brass corrodes pretty much in the sulfur compounds that come with burning black powder, and after about a century of so it just all turned green. Pretty much know where it is in the basement, if I find it while this thread is still alive I will post it.
Yes, yes, YES, I know I am a wet blanket.
If you forge up a tube which someone might be expected to shoot you have a moral responsibility to proof test it, twice, just like the Armory did. The "bang" is followed by a close inspection for any defects that may have opened up in the bore.