Flattery will get you everywhere.
You've done quite a few things I've never tried, so I don't have a lot of confidence when it comes to giving you advice. But this is what I would try if I was going to start over. Hold in mind that whenever you do a restore there is a chance you will loose something, so be double sure you've got everything you care about saved off the iPad.
Open iTunes, go to Preferences, the Devices tab, and prevent auto sync. This will prevent unwanted backups and syncing. It also prevents auto-launching of iTunes, so you may want to turn it back on later, if you like that feature.
Connect to iTunes and do a Transfer Purchases, then a backup. Both of these can be done by right clicking on the iPad under the Devices list. Then restore from the Summary page.
When the restore is done you will have to re-initiallize the iPad in iTunes.
This is where you make a choice between a clean start, and getting your stuff back. If you want your stuff back, then choose Restore from Backup when you get the choice. If you don't get the choice, then choose it by right clicking on the iPad and selecting it. When done sync the iPad to get back your app, music, whatever.
Hold in mind that the restore from backup does not restore your apps, only the content of the apps. If after syncing the apps, if they are still missing content or game save, you can try another restore from backup. It's been known to work a second time, but not always.
If you want a clean start, go through your sync settings to make sure what you want is selected, then sync. This will probably take a long time, depending on what all you have.
For a supper clean start, and the longest recovery time, do not sync your apps. Leave apps syncing on, so that you continue to get them backed up, but uncheck them all. Don't delete them, because you may lose your access to discontinued apps that you want. Instead manually download the apps to the iPad straight from the app store.
Anyway, that is what I would try if I were in your shoes and had given up on all lesser remedies. No guarantees. Much of my info is simply what I remember from what I've read. There is no way I've got the resources or time to try it all out myself.