I posted this comment earlier and, when I re-read it, I noted that my typos made it hard to read. So, here goes the comment again, and I hope it is helpful if you decide to expand the story.
I feel you need to show how Tom Bertram's character changes as he reflects upon his former behavior. Austen writes that illness made him more thoughtful. What did he learned as he mused upon the man he had been and the man he would like to become?
Here are some past events that might trouble a more thoughtful and set him trying to figure out amends:
As a teenage boy, he gave the prepubescent Fanny essentially the same off-hand casual gift all the time. What if the adult Tom gives a Fanny a gift that shows he now understands her heart better? She is touched and starts to see him in a different light, not just the cousin whom her good heart would always insist she care for, but as a man who stirs her?
Fanny's first loyalty has always been to Edmund because he made her his friend/student, and took her education in hand, while Tom mostly ignored her. Perhaps Tom and Fanny could have a discussion about that? There comes a time to leave off with childish things, to become a woman in your own right rather than in someone else's image. Fanny could come to believe she might be the woman with Tom that she could never be with Edmund.
As a young blade about London, and the eldest son, Tom overspent and put his father into a bind. The only way it could be corrected was to slice the inheritance that would have been Edmund's. Sir Thomas chose to sell a living (a clergical appointment) in order to raise funds to pay off Tom's debt. The appointee would own the living until he died or resigned it willingly. The loss to Edmund in future income could have have been quite severe as he waited for that living to come back into the possession of his family so that it could be given to him.Tom tried to excuse himself at the time that appointee at the Mansfield Park living, Dr. Grant, was fat and would no doubt quickly kill himself overeating. This was another casually cruel assumption by Tom, and a better Tom would have to recognize this. I am not sure of what scene(s) would take place but Tom's seeing his error in this would mark further heightening of his character.
In canon, Tom was the one who pushed for the also ill-conceived theatricals, shrugging off what his father would think. He knew his father would not like it and, hence, he was shaking in his shoes when Sir Thomas returned home unexpectedly. It is clear in canon, however, that Sir Thomas knew his sons well enough to expect it should have been Edmund who stopped the foolishness. He knew that although Tom was the eldest, he was also the screw-up who could always be expected to do the wrong thing. Tom needs to reflect upon his disobedience and failings as a son, particularly the eldest son. He should be shepherding his siblings rather than merrily leading them astray and hoping they can "get away with it" before Daddy comes home.
Also, although Tom was the scion who would inherit the family title and money, the ambitious and cynical Mary Crawford chose to pass him up and to seek Edmund instead. Her choice tells you something about the two men. Mary Crawford was not a good woman or a particularly wise one, but she was smart. (Losing Edmund might even have made her wiser.) The point is that this smart and cynical woman saw Tom as a second choice at best, if even that. That is more evidence that Tom needs to make himself into a better man, someone to be depended upon, someone who can lead a family to prosperity and maintain solidity.
Finally, a romance between Tom and Fanny cannot be solely based upon our anger as readers at Edmund for being slow in recognizing that Mary Crawford was not the woman, or even "a" woman, he should be consorting with. We cannot make Tom good by simply acknowledging Edmund has flaws. I don't think that a story in which Tom and Fanny get together should trash Edmund because I think would always care for him, even if another man managed to steal her heart.
Again, I hope these thoughts, or, at least some of them(!) are helpful, and I hope you will write more.
