Part number 0045418601, "electronic consumer" battery in trunk, the stealership wants $250 for it, there must be a cheaper solution does it really need the factory battery???

 

The battery is an absobed glas-mat type and they are expensive.
Have a word with your dealer and see if he'll do you a trade discount.

I replaced mine last year and here in the UK (where we usually pay a LOT more for everything) it was 111 pounds from my MB dealer - I would expect that to be $111 in the US. (usually we pay the same in pounds (or more) as you do in dollars - I'm surprised it is so expensive for you).