Which integration pattern should be chosen to display a customer's billing history without storing it in Salesforce?

The integration pattern that is most suitable for displaying a customer's billing history without storing it in Salesforce is Remote Process Invocation - Request and Reply. This pattern is designed for scenarios where a real-time request is made to an external system that then provides immediate data in response.

In this case, when the customer requests their billing history, Salesforce can make a direct call to the external billing system to retrieve the relevant information. The data is fetched dynamically and used to display the customer's billing history on the Salesforce interface without the need to store it in Salesforce. This allows for up-to-date information to be presented while maintaining data integrity and avoiding unnecessary data duplication.

In contrast, the other integration patterns may not meet the requirements as effectively. For instance, Batch Data Synchronization is intended for scenarios where data needs to be periodically synchronized, which conflicts with the need to display live data without storage. UI Update Based on Data Changes focuses on updating the user interface in response to changes in data already stored in Salesforce, thus not applicable since there's no intention to store the billing history. Remote Call-In involves direct requests to Salesforce from external systems, but here the need is for Salesforce to reach out to an external system, which is better suited to the Request and Reply approach.