How do you access under a front loading washing machine drum to remove a trapped coin?
There is certainly a method for removing/replacing the drum.  Most newer models have exploded parts diagrams online, both by the manufacturer and third party appliance part vendors.  The vendors frequently have videos (hosted on YouTube). You can google them by using the washers model number.

As far as fixing the rim damage, one of the rotary tools, like Dremel, would make short work of making it smooth. Start with a metal grinding bit and finish with buffing wheels loaded with buffing compounds.

The dented section (above the rim in your picture) should not cause clothing damage, it just ugly.  

I doubt the drum will rust, as its likely stainless steel throughout.  Any paint damage (not seen in picture) that is scratched through the topcoat, should be primed and painted with automotive or appliance grade products.

I hope you get satisfaction from the shop.  This damage was careless and unnecessary. 

Do know that a full drum removal would have been quite expensive, due to the number of labor hours needed