"Allow Deletion of Running Applications" Option
The answer turned out to be something silly. I was using an app called "AppCleaner" to uninstall the application in question. The option to "Protect Running Apps" was actually in AppCleaner, not in the Mac OS itself. In order to delete the running application, I had to uncheck "Protect Running Apps." I was able to locate the Preferences within AppCleaner and re-check "Protect Running Apps."