The joplin-appimage AUR package and the AppImage in your ~/Applications/ folder are two different things.
When you run an AppImage and choose to Integrate and run, AppImageLauncher moves it to ~/Applications and copies a desktop file to ~/.local/share/applications/. To run it from then onward, use the menu shortcut. There are options in the menu context menu to remove and update it (when applicable).
This is not an issue with AppImageLauncher, it’s doing what’ it’s programmed to do. The joplin-appimage AUR package installs the AppImage to a read-only location, which AppImageLauncher is not meant to be used with. It’s badly packaged and I do not recommend using it.