While not ideal, an alternative approach here could be to setup your Azure database as a subscriber to replication publications created from your on-premise database(s). Replication will allow you to limit the articles within the publication, filter the types of records, etc. which hopefully means you can minimize the publications to only send the data necessary to serve the traditional four-part queries. If you still want to utilize cross-database queries, you can always create a second Azure DB which would act as the subscriber to these publications and then utilize cross-database queries from your current Azure DB which may allow you to minimize any rewrites of existing functionality.

While not ideal, an alternative approach here could be to setup your Azure database as a subscriber to replication publications created from your on-premise database(s). Replication will allow you to limit the articles within the publication, filter the types of records, etc. which hopefully means you can minimize the publications to only send the data necessary to serve the traditional four-part queries. If you still want to utilize cross-database queries, you can always create a second Azure DB which would act as the subscriber to these publications and then utilize cross-database queries from your current Azure DB which may allow you to minimize any rewrites of existing functionality.