there are too many "forward" in your question...
If I understand well, you are configuring APM as SAML SP authenticating against 2 different SAML IdP based on landing URI (first URI requested when not authenticated yet)
So Authentication is configured as expected, then you are requesting here how to configure SSO with back end server...
When authenticating with external SAML IdP, APM doesn't know the user password. So only passwordless SSO methods are available.
As Kees already answered, the best SSO method is kerberos. easy to configure and available for most of web services (Windows IIS, Apache, tomcat, ...).