On the BMI thing, sure, there may be some people with really odd physiques. There may on the other hand be far more people who justify their weight with such arguments.
The reality is that *if* you want to go significantly faster up hill, for virtually everyone except people already really thin, or the very unfit, losing weight is necessary.
The OP according to their own figures, has a BMI slightly north of 23. Froome's is 19.
That's not at all to say there aren't other ways to improve - I suggested a couple. But if already at some reasonable level of fitness, as being able to ride the described hill suggests, it's a near certainty it will make more difference than anything else up hill.
Of course, you can, like me, just accept your limitations and enjoy lugging your carcass up hills, swearing to refuse all cake in future whilst simultaneously looking forward to pudding tonight.