what is the cost of battery service?

In the US (these are internatioal English-language forums) the cost is US$199 plus any applicable tax and shipping. However, prices onsome item have changes so this Apple page should give you an accurate estimate:

Mac Service & Repair - Apple Support

Hi,

Apple offer Battery Service Program. Just wonder if your MacBook Pro suits this program.

13-inch MacBook Pro (non Touch Bar) Battery Replacement Program - Apple Support

Battery replace cost would be $199US.