I don't see much of a dilemma for Brienne as she has seen that Arya is more than able to protect herself.
Maybe. However, Brienne treats this as an absolute commitment. She didn't pledge to "Protect Caitlyn's daughters, if they seem to need protection."
I could see an opposite scenario: After much deliberation, Brienne decides that Sansa is surrounded by a strong fortress and thousands of loyal troops, while Arya is about to go out into the wild, alone. Arya declines Brienne's offer of protection - after all, it's hard to sneak around King's Landing when you're in the company of an Amazon in bright armor.