I doubt you could weld, solder or braze the pipe together. They're usually quite thin and it would take a fair bit of skill to do it and maintain the nice chromed finish.
I'd be very shocked to find that these threads are actually bespoke to the manufacturer (or even fixture) in question. Most things are made with standard threading because that's the easiest and least expensive thing for the manufacturer to do. Using bespoke threading means developing bespoke machinery to make the threads and that costs more in manufacturing. I'm sure this cost plenty to purchase, but if it had bespoke threading, it would have cost even more.
Honestly, I would look at purchasing a whole new shower system and scavenging a single, appropriately sized piece of pipe out of it. The key factor for the new shower head setup is that it has a finished pipe with the correct, bright chrome finish that has a piece of pipe with threaded fittings at both ends (I presume one male, one female) and is roughly the 6" you need.
The other option would be to find one with a straight piece of piping that is 6" longer than the vertical piece you've got and replace the whole thing from the mixing valve at the bottom up to the split right below the upper mounting bracket.
No matter how much the new shower head setup costs, it would be less than tearing out all this plumbing & tiling to move the whole thing up 6".