For anyone who find this Thread in search for a solution, you don’t need to uninstall “The Final Hours Of Mass Effect 3” at all if you don’t want to.
Close Origin then go to your: (Disk):\Program Files [(x86)]\The Final Hours Of Mass Effect 3\TheFinalHoursOfMassEffect3\BIOGame” folder.
If you already downloaded the DLC’s there should be a “DLC” folder, relocate this folder (integrate if you have other DLC’s already installed) into the: (Disk):\Program Files [(x86)]\Origin Games\Mass Effect 3\BIOGame” folder.
Start Origin and try the installation of the DLC’s in Origin again.
Should this not work, rename the “The Final Hours Of Mass Effect 3” folder in "_The Final Hours Of Mass Effect 3” then restart Origin and download the DLC’s again.
Don’t forget to rename the “_The Final Hours Of Mass Effect 3” in “The Final Hours Of Mass Effect 3” after the DLC installation.