From my research Continental is the only supplier that does 235/40R20 tires and these are definitely not a winter tire. You might have to consider going to a smaller diameter wheel to get a winter tire but make sure it clears the brake calipers. The CLA 45 & A 45 use 19 inch rims as standard so this might work.
If Tire Rack says a 245-45-20 size tire will work, then it will work. I know they always get a donor vehicle and test the sizes they sell. I have 245-45-19's on my GLA, which is a 'tiny bit' taller than OEM, with zero issues.
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