]It is merely a supposition, but perhaps JA uses the word 'obey' to highlight again the almost father/daughter like relationship that exists between Darcy and Georgiana. Rather like, a child's obligation to obey her father, somewhat the same applies here between brother and sister.

This is what I thought, too. Georgiana isn't used to issuing invitations, as the passage notes, and I can see her simply being one step behind in what etiquette dictates she do next. Then once her brother pointed it out to her, she "obeyed" because she knew him to be correct through his greater experience. I'm sure it's meant to be taken as "willful obedience." ;)