That's a cute little disc. I had one the same size that went on a 1950's Simplicity walk behind tractor that originally had a 3hp motor on it and no doors. Times change.
I've used a hi lift jack, the out rigger on a backhoe, bucket on a loader and several other methods up to and including a sawzall. For the stuff under 17-18", Earl across the road has a commercial tire machine that usually works if we sweet talk it. On the bigger stuff, farm tires and such, my favorite way was to call Terry the Tire Man and $10 later they were fixed. Terry is almost in a wheel chair last I heard. So I do what I can. The only large tire shop in town wants you to bring the tire to them. Ever hump even a small tractor tire, say a 14.9-28, into the back of an F350 the previous owner put a lift kit in? Fun. It's all filthy, dirty, hard work even with the right tools.
Not sure what's on the schedule today. Moved the sheep to fresh graze last evening, gotta wait for Mon to get one last bearing for the round baler, wife and girls are going to the movies this afternoon. Still too mad at the pastor and politics to go to church. Might work on the skid steer and surely I'll knock down hay if the sun comes out.