Characterization is a classic fanfiction pitfall, and Austen sequel readers are especially picky about this. They know these characters inside and out, and they want to recognize their favorites in our stories. To avoid their ire, we work hard to present the characters exactly as Austen left them.
However, that's not true to life and it's not good writing. People grow and change; I will not be the same person at the end of this year as I was at the beginning. That change is even more vital in fiction. If we are to believe the crisis that drives your plot matters, your character must be changed by the events.
In Charlotte Collins, Jennifer Becton strikes the middle ground. Becton begins her story with the Charlotte we all know, the lady who believed that "happiness in marriage is entirely a matter of chance." If anything, after seven years of marriage to Mr. Collins she is even more pragmatic than before.
After her husband's death, she is assiduously courted by two men; one a gentleman, the other a rake. When the rake ruins her reputation by spreading a scandalous falsehood about her, she is faced with a decision. Will she remain the same Charlotte who chose security over all other concerns, or will she risk everything for one chance at happiness?
Our last picture of Charlotte is of a woman who has finally found true love. Her life is no longer ruled by propriety or a striving for security. Her life is exactly the opposite of where Jane Austen left her, but those same purists we fear have done nothing but praise this book.
There are three keys to Becton's success. I already touched on the first--Charlotte starts out as the recognizable Austen character. The second is an inciting incident that is both believable and big enough to cause the ripple effect you're looking for. Mr. Collins' death certainly qualifies on both counts.
The third is where the magic is. Charlotte doesn't transform overnight. This is where that little thing called plot comes into play. Each decision Charlotte makes drives her to the next change, which leads to the next decision. When we finally reach the climax of the story, she is sufficiently changed that we can almost guess what she will do.
Character development is a difficult balancing act when writing a Jane Austen sequel, but if we pull it off, the result is a book that will satisfy both the purist and someone only looking for a good story.