SOLUTION
Tested on MacOS 12.6
Hi there,
1) Fully uninstall ASC and Arturia VST/Plugin
2) Install ASC to the standard "Library/Arturia" and log in to your account then close the app
3) Open "System Preferences" -> "Security & Privacy" -> "Privacy" -> "Full Disk Access". Now unlock mod in the lower left corner and allow "Arturia Software Center" & "ArturiaSoftwareCenterAgent"
4) Open ASC and change the resources folder as you would normally do to your favourite location. This time, the ASC won't crash and ask again for your credentials.
5) In the ASC install all the VST & plugin
6) Some VST could eventually pause the download at 100% so don't close the ASC or try to replay download button
7) Open "System Preferences" -> "Security & Privacy" -> "Privacy" -> "Full Disk Access". Unlock again the mod button. This time, allow "com.Arturia.InstallHelper". Then manually allow VST by clicking on the "+" button and browsing to the App location eg. "Applications/Arturia/Analog Lab V".
After having manually allowed all Arturia VST, go to the already opened ASC and click on the Paused "Downloading"
9) Everything now will completely be installed on the external SSD or whatever you choose
10) Before testing any VST/plugin, go to "System Preferences" -> "Security & Privacy" -> "Privacy" -> "Full Disk Access". In the list, you will see a duplicate of every VST you installed. The former will result already allowed in the previous step, the latter requires to be allowed on this step. Allow those remaining duplicates and close "System Preferences".
11) ENJOY
The screens herewith attached show what needs to be allowed to have everything properly working!