I've used a lot of the Brownells 44 solder. I just use common No-Corrode paste soldering flux with it and it works fine. I've re-ribbed a lot of SxS bbls with it for years that way.
One draw back is that the solder 'line' will stay bright white and not turn dark grey/black like common lead/tin solder will after a while.
One reason I've gone back to 60/40. Plus the slightly lower heat needed.
Go to the home improvement store plumbing section and look over the solder selection. A good portion of it is now 'lead free' (soft) solder. Different % alloys of tin and silver. You can probably buy what you want right there as well as 50/50 or 60/40 lead/tin.
It's all usable with a common propane torch. 300/400 range +/-
If you use any of the 'acid' flux like comet flux and the like, yes you will have after rust as they are chloride mixtures. Mainly zinc chloride and ammonium chloride (sal-ammoniac).
Some have hydrogen chloride in them. Great chemicals to brown bbls for you!
Stay out of the fumes....
No real acids,,just alot of chemicals in the Ph acidic range.
You have to neutralize them or the metal will keep on rusting.
A real problem w/ SxS bbls if someone puts them together with one of the acid flux mixtures. They'll keep rust inside,, and sometimes out.
Even touch up jobs using an acid flux which is commonly done 'cause the stuff works so @!*% well, causes problems with after rust.
I have a Win21 in here now that the customer found rusty looking moisture running out from under the forend.
The problem,,the forend lug was resoldered with and acid flux. The bbls then polished and went through re-rust bluing process.,,and still the acid flux remained and rusted that area.
It also gathers moisture from the air like a bad flashlight battery and it then dribbles out and over the area. It takes the blueing off as you can imagine.