You wouldn't be referring to this brake would you:
Some have gotten not-so-square bends and it seems the reason is the brake is just light-weight enough to require reinforcement.
Others have tried this with Tapco flats and results are mixed. General consensus is that it isn't quite up to the job and you need to buy some bar stock from HD to reinforce it. Since you're talking about welding, that shouldn't be a problem. True, you are practically rebuilding it, so the $21 gets you the precision-milled flat surfaces and properly aligned hinge - still a good bargain.
I've read that gratuitous use of C-Clamps help too.
I have picked up one of these but have not tested it yet.
It seems that you would be better served by using 1.25" bar stock and doing the SKS method of bending (via a vise). A shop vise DOES have the capability of bending flats. You just need to go slow and grease it, and be careful.
FWIW, I know that *I* myself could not weld it properly, I haven't the welding skills. I'm just a crazy guy with a garage full of toys, er, tools.
To address your question - you COULD weld it I think. If there were a way to minimize and/or recover the metal after flexure, then it would be doable. The chances for failure are high - the sheet metal will undoubtedly suffer a great deal of heat flexure that I don't think can be avoided. That's why silver solder won't work for welding trunions, and the laser-cut weld-up flats are such a bugger to use.
BUT it would be no worse than using a weld-up flat. I think the key would be a huge chunk of copper to absorb the heat, and doing the spot-weld alternating on both ends, letting it cool, another spot weld, etc. etc. I remember reading that method on one of those "how I welded my laser cut flat" posts somewhere. Someone with better welding skills can better explain this (WinnR knows someone).