Let's break this down.

The sentence contrasts her reaction to a mere peccadillo (small fault) with her reaction to far more untenable behavior.
She condones the worse behavior with only a "wagging finger" (mild scolding), so for the small fault she must react more strongly - but in what way?

The structure:
- "Though she would _______ at a mere peccadillo, she would ... condone far more untenable behavior."
- The blank must be a verb meaning to show annoyance or offense at something minor.

Option analysis:
- bridle - to show irritation or resentment $\(\rightarrow\)$ fits
- quibble - to argue over trivialities $\(\rightarrow\)$ could fit, but less about taking offense, more about petty argument
- bristle - to react with anger/offense $\(\rightarrow\)$ fits well
- beam - to smile happily $\(\rightarrow\)$ opposite meaning
- chortle - to laugh $\(\rightarrow\)$ not fitting
- guffaw - to laugh loudly $\(\rightarrow\)$ not fitting

We need two answers that are similar in meaning and fit the context of being offended/annoyed at a small fault.

Bridle and bristle are synonyms in this sense (both mean to show irritation).