In Pride and Prejudice by Jane Austen, there is no inherent connection between independence of mind and women's happiness. Instead, happiness seems mainly due to there being a good fit between taste, temperament, and circumstances.
Elizabeth Bennett has an independent mind, and rather improbably ends up happily married to Mr. Darcy who is not only wealthy but honorable and kind. Although this is a happy ending, and most readers find Elizabeth a sympathetic character and are glad she ends up happily married, the reality of how marriages worked in the period suggests that this independence of mind and unwillingness to settle for less than a perfect marriage would not have worked out well in reality.
Jane is a more pliable character who also ends up getting happily married. Unlike Elizabeth, she has fewer conflicts with the society around her and is more conventional; this seems an asset in terms of her future happiness.
Charlotte Lucas is a clear-headed and pragmatic character, lacking Elizabeth's idealism, and perhaps the character most likely to thrive in any circumstances. She accepts that she has few choices in life, and is determined to make the best of her lot, accepting the realities of the world in which she lives. Her low expectations and willingness to settle for a basic level of comfort means that she is content in situations that might make Elizabeth or a more idealistic and independent woman miserable.