Hello,
I do not know that plugin but from what I’ve read about it I do not recommend using it on the staging site at all.
It seems the plugin can not differentiate between your production and your staging site so it may happen that you add data from your staging site to google that you do not want to.
In my opinion, there is absolutely no reason to send any data from your staging website to a search engine. The staging website should be excluded completely being listed in google. You can not guarantee this by using the Google Site Kit on your dev site.
Cheers
René