Execute SSIS procedure from stored procedure using non-privileged SQL login
As far as I know session started with SQL Server authentication will not let you create security context outside of SQL Server (because the only external context available is SQL Server service account and EXECUTE AS will not let you impersonate SQL Server service account for external access).
SSISDB procedures has to have context outside of SQL Server, so session has to be started using Windows Auth.
You have few options, but all needs more changes that you mentioned in the email:
1. I think easiest: create SQL Agent job wrapper.
2. Create SQLCLR stored procedure creating new callback session (it will use service account = sysadmin, I would say too privileged).
3. Enable Service broker service and use activation to start SSIS.
4. Publish SSIS package as a view and let your app consume the view.
5. I didn't try this, but worth trying to use a loopback linked server.