Hi. Any rifle will do for varmints. The only thing you'd have to worry about is where in Southern Ontario you're shooting. There are municipalities that don't allow anything greater than .275 calibre, some say .270. Both by the cartridge name, not the bullet diameter. As daft as it is.
It's not all of Southern Ontario either. Mostly in Southwestern Ontario and around TO, but you have to know. You'd have to know about any firearm discharge by-laws too. Most municipalities have their by-laws on-line or you can call the County/Township office during business hours and ask.
A 6.5 x 55 Swedish Mauser is ok. Not exactly cheap shooting and there's little or no milsurp ammo, but the rifles are accurate. The 6.5 will do nicely for deer, bear and moose too.
"...with my M1..." Done it on ground hogs. Astounding what a 220 Silvertip will do to a ground hog. You likely don't want to feed your rifle a steady diet of 220's though.
"...taken with a trappers licence not a hunting licence..." Depends on where you are. Small game hunting licence in Ontario. Some provinces say no FMJ's too. Mind you, commercial FMJ's aren't the same as military FMJ's.
"...live fire training..." Know a guy who was on a BREN range, one day, when a rabbit hopped out onto a down range shooting position. Five instructors left a big hole in the berm.
"...who buys them?..." Some of the fur sales places will. Or have the hide(s) tanned at a taxidermists.