We ended up changing our code to execute and download the report using HTTPClient method using the HTTPS URL of the report.

This link Microsoft explains how it works.  Attached is...

No, we are using SQL 2016 and 2019.   The ReportExecution2005.asmx is the end point to call the SSRS web service as outlined at https://learn.microsoft.com/en-us/sql/reporting-services/report-server-web-service/methods/report-server-web-service-endpoints?view=sql-server-ver16.

We have a web application...

Yeah.  Our DB server is separate than our App Server (IIS).   DB server has 32 GB of memory.

We use our own desktops for development with Visual Studio.

I'll try running after...

You said something about Azure made me think of Azure Data Studio.  We don't have any PROD databases in the Cloud but I was playing around Azure Data Studio recently...

These databases are onsite.  I'm using the same SA username and passwords for other databases on that SQL instance.   I don't think it's related to security.