Since you want to execute the sub-process as a background activity, would it be possible to do this by using the Invoke 2.0 REST API call?
See if these links help generate ideas:
REST Invoke API - DevTest Solutions - 9.5 - CA Technologies Documentation
https://support.ca.com/cadocs/0/CA%20DevTest%20Solutions%208%200-ENU/Bookshelf_Files/Invoke/index.html
or on your DevTest installation, navigate to: http://<yourDevTestServer>:1505/api/swagger/#!/CoordinatorServers/startTestActions_post_9
Are you passing any values from the process to the sub-process? If so, I do not believe there is a way to pass those values on the REST API call. In that case, you will need to place the values somewhere (perhaps on the Persistent Model Map or in a DB table) so your "subprocess" test case can access and use them. Your subprocess would be responsible for cleaning up after itself with regard to these passed values.