i dont think this would work because ignition is a momentary switch and you would have to hold your key in ignition mode or start and not just on. you would also have to push your push button switch while you hold your key in start.
someone correct me if i am wrong but i think that answers that question.
this is a guess but i think you might have to rewire ignition so it does not work with the key in ignition only. when it is in the "on" position power is supplied to all of the switched power accessories and whatnot and if you had a momentary switch or button installed in place of ignition the car would start when you push it...
so long story short i think you would have to cut the ignition wire and hook it up to a momentary switch instead of ignition>button>start like you said.
it would go from stock (key turned to on>power is supplied>key turned to ignition> start car)
to this (key turned to on>power is supplied>push button instead of turning key to ignition>start car)
i could make a wire diagram but i dont really want to sorry. and also this is just based on my knowledge of switches and electricity from looking around online and stuff, i dont have a push button start but that was my best guess, hope it helps