I have over 5k people who like my page. The group I created within the page only lets me invite one person at a time. When I click "invite people who like my page" the screen comes up but I'm not able to click on anything and no names appear, just an empty box where it appears names should be listed.
Yes there is a way to add people in Facebook group with script instead of adding them one by one manually. This method bypasses limit of 1000 members so now you can add 1000 plus members in your Facebook group.