Three stone is rather a lot to put on in a year, if you don't mind me saying so.
You're doctor may well have "flat-out refused" to believe your diagnosis of implanon, because that is totally implausible (I don't actually know exactly how implanon effects your weight, so maybe it is possible, but 3 stone is a hell of a lot given implanon doesn't actually involve a very high dose of hormones - it's progesterone only and at a dose less than most pills), especially because you are eating healthily and exercising.
I mean.. don't you want to find out if you have a thyroid problem or one of a host of other diseases that can cause the kind of weight gain you're talking about? What has made you dismiss the possibility of hypothyroidism?
I really sympathise, don't get me wrong, but don't presume you know more than your doctor (well, not always anyway). It would be irresponsible of her to remove your implant and just send you on your way - when that might be unnecessarily depriving you of contraception, and not finding a solution to the problem at all.
EDIT: Although, don't let your doctor ignore the problem, or blame it on a bad diet if you know that your eating habits haven't changed. Weight gain like that should be investigated, and if a thyroid test comes back negative it could still be something else.
And if you ultimately still want your implanon removed, that is your right.