Getting the leather the right shape :-
If you are starting with a stick with shaped padding and then stretching the leather over that padding, the leather should take the shape of what is beneath.
As for stretching it evenly, you could use the same principle that tensions a traditional drum head. Make holes in your leather, maybe four corners, or spread more evenly around a circumference of a circle. Soak the leather to make it pliable and stretchy. Thread cord through holes (possibly between pairs of holes on opposite sides of leather) and pull tight towards the opposite end of drum stick (maybe make something to fasten them to? a wooden plate? another piece of holed leather?) Adjust the strings so that leather is evenly tensioned, then tie binding around the base of the padded drumstick head. Once leather has dried and is stable, untie tensioners and trim leather to look right.
Quack!
GtD.