using irules to deliver pages

The description of your issue is confusing and better edit your question with precise description of the issue.

For the ASM VIP to APM VIP you use Layered VS (https://support.f5.com/csp/article/K54217479) ?

You can also configure the ASM to protect the APM login page, so no one can can click other pages if they have not passed the login page:

https://support.f5.com/csp/article/K13315545

I think that you can use the Per-Request policy to force again a login.

Better review your multy domain config if you think that there are issues as you need to be carefull with profile scope (https://devcentral.f5.com/s/question/0D51T00006j20Ce/v12-apm-profile-scope):

https://techdocs.f5.com/kb/en-us/products/big-ip_apm/manuals/product/apm-authentication-sso-13-1-0/27.html

https://clouddocs.f5.com/training/community/iam/html/archived/class7/module1/lab7.html

For to clear APM session with iRule use "ACCESS::session remove"

https://clouddocs.f5.com/api/irules/ACCESS__session.html