Blesta lists transactions in the interface by themselves, without invoice/service information. If a client wants to see what that transaction applied to, they click on it and another request is made that will determine to what invoices it applied the transaction, if any. Determining the service would be another step, but Blesta doesn't go any further to show that information.
I think the best work-around is to write a custom query that will fetch exactly what you want. You can create a plugin for Blesta that provides a method that fetches exactly the information you want, then you can call that method over the API.
