How can I solder/braze thin stainless steel wire?
Successful soldering (you're not welding, and you have the wrong torch if you want to weld) requires that the work be physically clean (scrub it) and chemically clean (flux) - you need a very aggressive flux for stainless, because the reason it IS "stainless" is that it forms a strong oxide layer which protects it from most corrosion. Kester is not going to regard stainless as electrical work, for one thing, so don't bark up that tree too long.

With thin stainless, the other problem you almost certainly have (stainless conducts heat poorly, which makes it very easy to overheat) is overheating the work - once work has been overheated, you have to return to physically cleaning it and refluxing before you have a hope in heck of getting solder on it. You also need to stay out of the "oxidizing cone" of the torch flame - ideally, switch to an oxy-acetlyene torch, make the flame rich, and only use the feather - but you can get there with propane if you're careful. The thickness of the solder does not matter much - you heat the wires, and melt the solder on the wires - don't heat the solder.

You could also move up to brazing, which is a process similar to soldering but at higher temperatures, using different filler metals. When most folks refer to silver soldering, what they actually mean is silver brazing (the rod has some silver, but not a huge amount.) The solder you linked to is a low-temperature solder that has silver in it, from the "lead free" change in "soft solders."

What this is doing on a DIY home improvement site is another question. High voltage electrical projects are not exactly the bread and butter of DIY. Another approach in a lab setting would be to do the whole job in a glovebox full of nitrogen or argon. You could also TIG weld it, if you were good at TIG welding. Spot welding might be another approach, but that will also work better on stainless in an oxygen-free atmosphere (glovebox, etc.)