Ouch. No comment on the shop, as it's obvious. electric trailer drum brakes are about as simple a system as it gets. Either there is power going to them or not and worst case, slap whole assemblies on all 4 corners. They're like $40-60 per wheel, pre assembled ALL components hide hair and feathers.
Use this info dispute any egregious charges.
1. Good wire from trailer plug back to the axles where it splits 4 ways to each wheel.
2. Good wire from there to each wheel.
3. Good magnet and properly adjusted at each wheel (or $34.99/wheel for a whole new pre-assembled 10" brake assembly including backing plate is almost a foregone conclusion if one needs to replace ANY of the components of a wheel brake.
Hope this helps with your ability to get fair and competent results with the shop.