Given that you have a have a Michelin all season tire that is in fact
on your car, and apparently works fine except for skidding on snow, I
don't think you have a snowball's chance in hell of getting the MB
dealer to pay for this.  There is a wide variation in performance
ability of different tires, especially with regard to snow or wet
surfaces.  Years ago I had Michellin MXVs on my 300SD and the rear
wheels would spin easily on acceleration on wet pavement, which shocked
me.  I went to Pirelli P4000's and they were significantly better.  Now
I have BF Goodrich Traction T/A and they are far superior in wet
performance than either of the other two.  I don't even notice them
ever slipping now.  BTW, this is just an example, as wet and snow
performance are two diff things.

To get the dealer to pay for it, you;d have to take them and/or
Michelin to court and prove it was their fault.  It's unlikely you
would win and the cost to try could equal or exceed the repair cost.
Hopefully you have collision insurance.