No. It's not possible to install FP6 and FP7 together in this way, because the setup program checks if there a previous Forest installed, and removes it.
This is done to avoid conflicts with files of different versions.
You could do this:
1.- Install FP6 with the command line. Since there is not FP6 for Max 2023, it will be installed only for 2021.
2.- Copy the files for FP7 manually, for any workstation.
In the latest updates, the plugin for Max 2023 is installed at C:\ProgramData\Autodesk\ApplicationPlugins, following the new Autodesk package format.
Basically you must copy the content of C:\ProgramData\Autodesk\ApplicationPlugins\ForestPackPro2023 to the render node, in the same location.
I hope that helps.