What would have to change to make such a story Persuasion plausible?
To make Persuasion a plausible story you may have to change the attitudes of the characters a bit. Although they are realistic characters who represent a reality of the time (the need to keep up with appearances, distinction of rank, and vanity of possessions), they seem to be too black and white to be as real as they can be. The characters seem to be "sold out" to their characteristics and appear rather inflexible.
Even Anne is much too stoic for my taste. All of us have a breaking point. No matter how strong and independent Anne is, I have a hard time thinking that she was no less than devastated when she realized the mistake she made when she lost Wentworth and when the family came to ruin. Especially a woman of her rank and with a need to marry in that type of society.
Hence, the characters, not so much the situations is what make the story a bit on the totally fictional side, and prompts us to realize that Austen made full use of her creative license to come up with these ideas.