From here
If the only copy of Jade Pearl Inn is in a province then isn't that an out of play area and therefore can't be named by the Shadow Estates.
Jade Pearl Inn & Shadow Estates of the Scorpion
A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
From here
If the only copy of Jade Pearl Inn is in a province then isn't that an out of play area and therefore can't be named by the Shadow Estates.
Please note his first sentence. The Scorpion player already has a Jade Pearl Inn in play.
If there was not a Jade Pearl Inn in play, you're correct, it wouldn't be able to be named using the stronghold.
On: Jade Pearl Inn