I re-read this chapter last night, and was also perplexed by the term 'obeyed'.

I think it is supposed to give us some insight into Miss Darcy's character but more importantly towards her complete and utter faith in the judgement of her brother.

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.

Also, 'obey' implies that Darcy expects complete agreement from Georgiana, as opposed to Georgiana wanting to give Darcy her full support. It seems to me, in light of what we know about her devotion to her brother the term 'obey' would be irrelevant because she would, as you say, 'readily agree' in any case.

Also, perhaps after the Wickham debacle Miss Darcy is more inclinded to follow her brother's lead more ardently than before?