Case offered an option called the "Flotation Kit" that consisted of the extra wide 12" tire/rim combo on the back end and extended axles with larger wheel bearings to hold those rims. It was known as the LL-120 Hi-Flotation Kit. The front rims and tires were the same no matter which rear rim size was selected. The flotation kit was an automatic option on the backhoe models. If you see a loader tractor with the flotation tires, then most likely it left the factor with a hoe on it and someone took it off years later. Rim diameter is only one issue. The height of the tire's sidewall also comes into play. This is very evident in car tires that come in profiles that are tall, such as a 70 Series and then get shorter with 60, 50 and 40 Series tires with a very short profile.
The difference between a 30 inch tall tire and 26 inch tall tire is 4 inches but you are only raising the rear of the tractor a mere 2 inches. That amount is not going to have a dramatic effect on how well your loader works because you can lay your bucket flat on the ground just by moving the control lever a small amount. The angle of curl won't be altered significantly either. If you are loading the bucket to the point of spilling material, the tire size won't matter because the bucket has no "auto levelling" feature built into it, either hydraulically or mechanically. You just load the bucket and then raise the loader arms until the bucket is level.
As I said previously, you have to think outside the parameters of the CCI world and seek out the AG wrecking yards. Lots of Ag equipment used AG bar tread tires. Phone around and ask. No need to tell them it's for a garden tractor. Who knows? You might get lucky and find an old combine with the correct 15 inch rims, ag tires and heavy wheel weights that would be perfect for your tractor.