Kinda hard to guess without a pic of the piston, but I would venture a lean conditiion- either an air-leak or a lean carb.
Inspect very carfully the carb boot and gaskets on that side, along with the crank seal. PTO crank seal may require pulling of primary clutch to inspect, MAG seal will require removal of recoil assy, and removal of flywheel and stator.
Go through the carbs very carefully, completely dis-assembling them, looking for any obvious blockage/debri. Blow out every passage with compressed air before reassembly. Check float levels. Inspect fuel lines for cracks/leaks, along with fuel pump pulse line. You may wanna take apart fuel pump and make sure that there is no crud partially plugging the outlet to the fuel line feeding the carb on the burned-down side. (I've seen stranger things happen!)
Like Jay said, EVERY time you are into the motor, run pre-mix in the gas. It is a cheap insurance policy against a no-oil burn down, and will help break in the motor. Once you are satisfied that you are showing oil usage (mark your oil bottle level, and monitor it for a couple hours of riding to see if the level drops), usually after two or three tanks of fuel, you can then rely on oil injection alone. This got me thinking- You might want to remove the oil line near the carb on the burned-down side, and make sure that it flows freely. Check the check-valve on that line as well, and make sure it is not stuck. I would think that of the two lines that feed each cylinder, if the one near the carb is plugged, it would more likely cause a lack-of-oil siezure of that piston.
Sometimes, if you are not knowledgeable... it may be as simple as finding somebody with enough experience to look at your sled, and go "Hey! look at this....." to find your problem. I've done it for guys over and over again. I know you are reaching out by asking us on the 4m, but it is much easier sometimes to find and pinpoint a problem in person. Don't be afraid to find someone and ask for help.
Post pics of the piston if you can. I'm curious.
