As I understand you need to get info about all API calls from RUM users sessions to your services? On Application you have list of services thar your aplication is calling. If you have separate service for API you can click on it, then click backtrace, pick only those calls that are starting from application and you should be good to go.
By application I mean Application in Dynatrace (where RUM data are collected). In services you have mixed api calls and other web server requests? If it is like so there should be option to split service by class / methods that are responsible for api calls I think.