Sims 4 external hard drive

@nyoochu  You can change the install location within Origin: hover over your username, select Application Settings, then Installs & Saves, and point Origin to a folder on your external drive.  You do need to create a folder, e.g. Origin Games or similar; Origin won't install directly on the drive, or "root directory."

Origin will still likely cache the downloaded files on your internal drive as it's installing, so you do need some space.  You can temporarily move other files to your external drive if necessary.  Downloading the base game alone, then adding the packs later, will also reduce the total amount of storage Origin tries to reserve on your internal.