As I said you probably won’t have luck with sqlite3. You could try to copy the entire sqlite3 package with all needed depdencies into your plugin’s folder. You need the sqlite DLLs somewhere as well, probably inside the Picard folder. But I have no idea if this will work or which obstacles you’ll encounter.
An alternative approach could be that you create a separate tool that handles accessing the sqlite database and which is a normal Python script running on your computer using the globally installed Python. Then you call this script as an executable from inside your plugin, and pass it the necessary paramaters.