While it's true that you can machine a pulley from a mandrel, on a lathe that small it will likely be a nightmare of fighting chatter. I'd suggest sticking with the face plate for maximum rigidity.
Be sure to check the face plate to see that it runs true, for if it doesn't, your pulley will have double the error, depending on how you mount it the second time. If the face plate doesn't run true, take a cut on it so it does.
If it's like the HF/Homier 7" minilathes, you'll find 3 or 4 nuts behind the flange that the chuck is bolted to. The chuck has studs extending through the flange. You'll find a number of holes through the rim of the flange when you pull the chuck off, number depends on lathe model, plus a raised register spigot in the center of the flange for chuck to center on. Mark flange and chuck before pulling the chuck so you can get it back in the same position. If it's like other
7xs, you can buy a faceplate for not a whole lot of money. Don't know about shipping to Canada, but HF has the faceplates as does Micromark. I've made one, it's not hard, the 6mm automotive studs came from the hardware store. Use hot-rolled steel or cast iron plate for material. Some guys have used barbell weight disks for material.
If you change chucks often, you might find the tool at
Some pictures of this same nut-starter appear in dropbox 2002 retired files* but the reference above includes a better description and picture of usage such that it takes less than 5 minutes to take off one chuck and get another in place ready to use.