HELP: 4-stroke engine oil mixed with gas in 2 stroke engine
Unfortunately the most likely failure point for bad oil is the piston rings.  This is a "get a new snowblower" tier problem. 

Here's what I would do.  Figure out how much fuel you mis-mixed (say 1 gallon).  Get a jerry can at least triple that size.  

Now, go make some new, additional premix fuel of twice that amount (e.g. 2 gallons) and mix the gas and oil correctly.  So you could just use that directly and be good 2 go.  

Now, drain the fuel out of the snow blower, and add it to this jerry can.  Mix thoroughly.  

Now, you have a can of 2-cycle gas that is 2/3 the correct oil and 1/3 not the right oil.  The "wrong" oil still has a fair amount of lubricity, and the 2/3 will cover other needs.  Go ahead and burn that, white smoke and all.   

This won't damage your engine, and is also the simplest way to dispose of this mis-mixed fuel.  You certainly shouldn't dump it on the ground, and mustn't dump it down a drain!