The do_settings_section() function call needs to correspond to the $optiongroup argument you pass to register_setting(). To see how all of the myriad functions fit together, see page 10 of my tutorial.
It does get fairly confusing trying to follow how the various functions string together.
EDIT:
You do appear to be using the option group properly. Can you clarify what is/is not “clicking” for you? You mention in the title that you want to use a tabbed interface, but the question text itself doesn’t really indicate where exactly you’re running into difficulty.