Hi Richard, In my (small and limited)experience, you should (SIF-) Bronze braze cast-iron pulleys that have cracked. Not sure about the "SIF" alloy, in case that has Cadmium in it, just use a "regular" brazing rod with the correct flux. (Flux the job well from cold and you get a really clean metal for the braze to "take" to).
Also for a steel shaft I would use stick - as I know some people are adverse to the slag, but professional welders get on with it and it really is a good solution to getting the right alloy rods and enough heat into the weld for good penetration. Or MIG.
One job I was on, we were MIG welding 1/2" welds in aluminium (joining 4" and 6" thick bars) and 5mm and 6mm rods onto cast steel components. - The welding Engineer had specified the size, material and current for the Rods (stick), not MIG, for the power needed for the size of weld penetration. I don't think TIG goes anywhere near that size of weld (Current), not that you will either. Slag removal should not be a problem, just time and care, which I know to be a good part of all your work.
Is it not easier to start with a new piece of steel and machine a new part?
Alternatively, instead of welding (are you just in need of repairing a gouged bit of shaft?) you may be able to braze fill the "bruise" cavities with the bronze - which should really minimise distortion compared to welding. How deep are the "bruises"?
A temporary repair may be a good quality epoxy "metal mender" (one of the filled resins, not clear liquid!), I am a bit vague... is the shaft just bruised so you need to fill less than 0.1" of cavity? Maybe a temporary repair will get the lathe fixed so you can use it to make a new shaft from scratch?
Or perhaps another "local" Engineer can make you one? ( I am in the UK, 8 miles from the original Washington, so just too far to drive to your Washington!).
Sorry I don't have any clever ideas, but Bronze brazing is my preferred method for many things, as it heats the whole job and minimises distortion on many repairs, especially on cast iron. Or you end-up heating the cast iron red hot before welding.... (and no advantage over brazing, except when repairing gear teeth).