The North American update has a Title ID of: 0004000E00021800
And the DLC for NA has a Title ID of: 0004008C00021800
You can search for those on hShop and it'll give you want you need. The main Plaza files are not stored on hShop, you can only get those from Nintendo's Update Servers. If hShop says you already have the file installed, then you have either the update or the DLC installed already.
Reinstalling it won't hurt anything. It also won't help if you aren't having issues.
The 3DS application gives you categories to search / load. I don't understand how you can't know what you've installed. Go Updates -> "Region" -> "application" (it's in alphabetical order by default). Press the Right Trigger to reverse the order and you can then press right on the D-Pad to scroll through whole pages. Find Streetpass Mii Plaza and install it that way.
Do the same for DLC -> "Region" -> "Application". If you have both installed, then you have the full Plaza with all the additional games.