I've not soldered directly to the barrel like this, but have soldered handmade pipes to a handmade rib on a half-stock TN rifle (rib was then screwed on). I filed the pipes carefully for a very good flush fit, then tinned both the pipes and the rib with a thin layer of low temp silver solder. Before attaching the pipes, I carefully filed away any excess that was in the wrong place. Clamped gently, then heated the pipes until the solder melted and joined the tinned surfaces. Two pipes. No solder showed.
I'm not an expert on this (or much of anything, for that matter), but it seems that keeping a thin layer of solder is important when you are tinning. When you put the two tinned surfaces together, you are doubling the amount of solder in the joint. So a very thin layer is sufficient. Anything beyond what is necessary becomes excess that then bleeds out of the joint.
_The Gunsmith's Manual_ (Stelle and Harrison) p. 175, when talking about soldering a rib to a barrel, even suggests wiping the excess solder off with a rag before it cools, presumably so as to leave a very thin layer of tinned solder before joining the parts. [Added: when soldering a rib, Stelle ahd Harrison do recommend filling the edges of such a rib with more solder as the piece settles into place under heat, and then clean up with a chisel-like tool.]
I found it helpful to work out the technique with soldering the pipes on scap material of the same size, using a prototype pipe I had left over. Trial and error. You may want to do the same on a chunk of scrap filed to the width of the barrel flat, etc.