ok, try this:
1. Uninstall all of them once again from Programs and Features in Control Panel
2. Restart your PC
3. Open C:\ProgramFiles for 32bit windows or ProgramFiles(x86) for 64bit windows
4. Uncheck all its hidden contents from Folder Option menu, and you'll see this folder: "InstallShield Installation Information" folder after you unhide them.
5. Delete all folders inside "InstallShield Installation Information" folder, then don't forget to hide back all of your hidden folders from Folder option menu
6. Click Start, type regedit in the search/run box, right click on regedit.exe, run as administrator
7. In Windows registry editor, open HKey_Current_User\Software\Electronic Arts, delete SIMS3 folder
8. In Windows registry editor, open HKey_Local_Machine\Software\WOW6432Node\Electronic Arts\EA Core\Installed Games, delete all SMS3 related folders (do not delete the rest other than SIMS3 if any)
9. In Windows registry editor, open HKey_Local_Machine\Software\WOW6432Node\Electronic Arts\Sims, delete all The Sims3 related folders (do not delete the rest other than The Sims3 if any)
(Note for 8 & 9: WOW6432Node is for windows 64bit only, you won't find it in Windows 32bit, but I forgot what its name in Windows 32bit, it's under "Windows" folder if I'm not wrong before you can find "EA Core" folder and "Sims" folder to delete its contents)
10. Close windows registry editor and restart your PC
11. Now try a new clean TS3 BG installation, run the BG from the launcher, after it's succeed, continue with its EPs/SPs you would like to add.
Hope that helps, I found that method also works to clean up TS3 corrupted installation/uninstallation