TL;DR:

Navigate to the steam install folder (example. C:\Program Files (x86)\Steam\steamapps\common\FINAL FANTASY XIV Online), delete the 'etc' folder, restart the launcher through Steam. Unfortunately as a side effect this might make the entire game re-download

For some context, I had previously downloaded the game as a free trial user and had converted to using Steam. While Mogstation showed that I was paying through Steam as a service account, I had never seen a prompt on my launcher to actually link to Steam. After some looking around i saw that the etc folder in the install directory had an install.def file which just had the text freetrial in it. I deleted the etc directory with the file, restarted the launcher, went through the account linking process and was successfully able to start the patch. Unfortunately as a side effect this might make the entire game re-download.