The path is stored to registry to HKLM (administrative install mode) or HKCU (non administrative install mode) to a subkey named after the AppId with _is1 suffix, stored under a key SOFTWARE\Microsoft\Windows\CurrentVersion\Uninstall (as you alraedy know). The value name is Inno Setup: App Path. The path is also stored to InstallLocation with additional trailing slash, as that's where Windows reads it from. But Inno Setup reads the first value.
For MSIX
Application User Model ID: 12030rocksdanister.LivelyWallpaper_97hta09mmv6hy
If Lively core is not running, launch it with screensaver flag.
For installer https://stackoverflow.com/questions/68990713/how-to-access-the-path-of-inno-setup-installed-program-from-outside-of-inno-setu
For MSIX Application User Model ID:
12030rocksdanister.LivelyWallpaper_97hta09mmv6hy