Battery cost

Currently in the US it's $49 plus sales tax if performed by Apple. Apple Authorized Service Providers can charge different prices, but many match Apple's prices. There will be an additional shipping charge if you choose mail-in service. There are different prices in other countries.

There are other other possibilities that could raise the price. If there's a crack in the screen, Apple or an AASP won't do the battery replacement without the customer paying for a screen replacement. If they open it up and see any of the liquid contact indicators show exposure to liquid, then they won't do the service and will offer a full replacement or will return the iPhone to the customer.