Edit - Ignore as i see you've already tried ~/Applications.
Edit 2 - Just tested under 13.5.2 (with a 'standard' user and an Appstore app) to make sure i wasn't going completely senile, as this was always the usual solution.
Copy (moving will create an alias/shortcut) the app to users ~/Applications (equates to /Users/username/Applications; create if it doesn't exist) and then delete out of the (systems) Applications folder; give Launchpad a few seconds/minute to update.