I have not tried using the method you are attempting, but have had good luck managing with AD Groups and a webservice, like this post: https://docs.recastsoftware.com/ConfigMgr-Docs/Community-Tools/Community-Tools-OneVinn-Tools.html#web-service-for-cm
As for your situation, you'd have to pause the TS, confirm it has the policy for those apps, and if it does, you could trigger the installs with PowerShell. I've never tried this idea, but as long as the machine has the policy for the applications, I don't see why it wouldn't work. You'd have to write a script that looked through WMI for all Application Deployments to the Computer, then start looping through and trigging the install. You'd want to build in error handling too.
There are several community tools that have similar code you could steal from, like CM Client Center or Application Tester
Application Tester: https://www.ephingadmin.com/test-configmgr-applications-automatically-with-powershell-and-hyper-v/
Client Center: https://github.com/rzander/sccmclictr
But I honestly haven't tested this idea, I would want to pause the TS after it has the client, and start playing around to see if it pulls down the machine policy (perhaps manually trigger that), then see if the apps policy pull down into WMI.
Good Luck!