Having only put one fowler together, (So I don't know what I'm talking about), I question why you would wait til the end to brown the barrel? Are you planning on unmounting the barrel after it is pinned to do this? I soldered my tenons on, then browned the barrel, mounted it and pinned it, then finished the stock. I was lucky enough to get the pins right the first time, let alone removing them and reinstalling.
as i said before , the barrel supports the forearm , the forearm does not support the barrel . So as your shaping , sanding , carving , it helps to have the barrel pinned to the stock . If the barrel is already finished , then you have to be especially careful damage the finish on the barrel .
At this stage the barrel is still going to come out and back any many times .
The order is also a little misleading as some of the things actually happen at the same time .
Take the carving . This should be done IMO once the stock is brought to the point its near ready to finish stain . But once the carving is done then you have some sanding and shaping to do up to the carving .
Also concerning the finish . IE stain and oil . While the stock is drying , you can finish the barrel . So once the stock is dry , the barrel is simply put back in the stock .
As to the pins . I leave them long until the stock has been brought to 90 + % shape . That way they are easily taken out and replaced