I have almost the same problem.
I bought Medieval when it was released 4 years ago. Then today I bought the deluxe since it's on sale and there's no other way to get the expansion. Trying to install the Limited Edition edition that came with the Deluxe results in a DirectX install error.
Here comes the fun part: After failing to install 3 times I hit Download on the normal Medieval box, it took a couple minutes to finish, and then it installed without any errors. Now both the normal and limited edition of Medieval says it's installed.and the download folder is gone, so I have no idea whether it installed the normal or the limited edition. Only thing is the Limited says there's an update available, but clicking the link does nothing.
Using the desktop shotcut Origin made to start the game skips the launcher (I assume because the website was taken down years ago?), which means there's literally no way to update the game.
EDIT: Uninstalled the game and clicked install on the Limited Edition cover. Game installed without errors. Pirates & Nobles also installed without complaining. It also no longer says that it needs to be patched.
Also re-read the automated receipt and realized that there is 0 difference between the normal and limited installers, since the limited content has to be unlocked inside the game. Now if I could just find the code for the extra throne rooms we got for preordering...