Microsoft's solution to this problem is the introduction of "Expression-based Connection Strings" (see this article for reference) which unfortunately does not apply to shared data sources.

Fortunately, by using a tool, some report design standards, and a Report Definition Customization Extension (RDCE), it is possible to change the Data Source References on the RDL file to point to the right Server-Database pair.

You can check this article:

Dynamically Pointing to Shared Data Sources on SQL Reporting Services using a Report Definition Customization Extension (RDCE)