Good evening, I've played this game too, and I think you have several difficulties compounding the problem.
Firstly the disc you bought had a tag in it on the periphery. This tells us it's likely the offcut from cutting a hole – so that's why it's cheap. Cheap it may have been, but you would have been much happier with a clean disc,
The plasma cutting action of itself (assuming it's just mild steel) won't harden the disc. That's not the problem, the issue is that the oxide layer (and the snotty grotty bits) left on the outside is (b$$$$y) awful stuff, and your poor little lathe did well to tackle it. Buying a blank which is clean cut on the outside will make your life infinitely easier.
That said, cleaning up the oxide layer with an HSS tool is a recipe for going nowhere. You need a carbide cutting edge of some kind, and inserted tools are good but there ain't no substitute for horsepower. Sorry, but that's cold comfort in your case. For some reason brazed tools are often better at the brutal business of getting back to clean metal, it's all in the composition of the carbide edge. That is as much as anything about the cost per edge of an inserted cutter against the re-sharpenability of a brazed cutter.
You can do an awful lot to make it easier by tackling the periphery with a grinder of some description – a bench grinder is good, an angle grinder with a linishing disc (sanding disc – but coarse and aggressive and a hard abrasive) – is better if only because it saves ruining the shape of the grinding wheel. A belt sander will do a good job.
Essentially the moral of the story is – as ever – you gets what you pays for. If you can find a supplier who DOESN'T stick you with a in-cut on the edge (where the plasma pierced the sheet metal before setting off on the circular path) you will get on better, a few minutes work with some sort of grinder will save you a whole bag of grief, and then get in there with a tool which won't burn out (even HSS will) if you approach the edge at a ridiculously high surface speed. If I buy a cut blank I expect it to be clean edges and would get very chopsy if it had a cut in the edge I had to machine away.
Good luck,
All the best Simon