How can I reinstall an application that the Mac App Store thinks is installed already?
Awesome, I figured it out. Run this command, which rebuilds LaunchServices:

/System/Library/Frameworks/CoreServices.framework/Frameworks/LaunchServices.framework/Support/lsregister -kill -r -domain local -domain system -domain user


And everything should be back to good! Immediately after running, the App Store allowed me to re-purchase the app.

Source: MacLife Link