There is a mechanism in the multi-site page that will detect the database needs to be upgraded.
The click the button to upgrade the database.
The last time I tried to use it the feature was broken. To get it to work, you have to manually set the site ID for the site that you want to upgrade. Because when you click the button you will see the system display site not found. Which if you look at the URL it is trying to hit the incorrect site ID. Help the system out by going to the site to be upgraded manually. In the browser type the URL of the site which you are trying to upgrade.
Is this the issue you are facing?
I have refactored the multi site code to fix the limitation and remove the feature touching those God codes in the system. I have been testing it for the last 6 months. I have worked out most of the bugs. I hope to release it by the end of the year.