Yes! Do check the transmission cooler. It's cheap and easy.
But the quickly oiling of the airfilter makes me wonder if you might have more than normal combustion gas blowing past the rings into the crankcase, carrying oil mist through its normal vent into the intake. Hence the mechanic's suggestion for a compression test.
It sounds like the mechanic you had look at the engine knew his stuff.
Here's hoping you have a simple problem we just can't think of yet!
Somewhere around Eleuthera, Bahamas