The Stinger, having a longer case, won't chamber in that short, match chamber. You'll get out-of-battery discharges, if it shoots at all.
Stinger isn't "crap", it's good ammo, it's just not compatible with a match 22LR chamber. I used to shoot a lot of it through my Marlin, and it was fine through that (not a match chamber, so I had no problems).
To figure out what's the "best" ammo for your barrel, you're going to need to buy some various ones and try them. Wolf MT seems to shoot well in a lot, Federal Automatch, CCI SV.
Look over on Rimfirecentral and see what people are using to win the postal matches and such.
I know the barrel you've got is a top quality accurate setup. You'll find something that will shoot VERY well if you do some testing. I wouldn't even attempt the Stingers, they won't chamber fully. The case is .090" longer than 22LR specs, and Kidd doesn't leave that kind of "slop" in the chamber.