The main interest is,“Can you use flying kick with ascetic form and your weapon?” But this applies to other style strikes too that call for a specific attack
You can use the chosen melee weapon with any class ability that can be used with an unarmed strike, such as an unchained monk’s style strike ability.
The attack made after the movement must be a kick.
So does ascetic form allow me to use my weapon there instead of a kick? Some style strikes allow any unarmed attack, but many limit you to specific unarmed attacks
Back when the feat was released, every single style strike in the game required a specific body part, and yet Ascetic Form explicitly mentions that it allows style strikes - indeed as the only example. In order for this to be true, i.e. in order for the feat to do what it outright says it does, it must allow you to ignore the body part requirements.
On the WMH's product page, all the talk on the topic was that the works like that, and the Author of the feat posted on the same page, making multiply replies, but did not correct anyone on this topic.
A year after the feat was released we got style strikes that do not specify a body part, but I don't see why the printing of a later book should somehow prevent the feat from doing what it was explicitly written to do.