Observational input..
A push button start is more than just a button you push. It senses engine RPM to know when to disengage the starter. It's electronic in nature, as well as mechanical. Just like a remote starter, it'd have to have safety protocols to prevent starting in drive, etc. On some newer cars, with a starter button, there's a sensor to know when the fob is inside the car, and the starter function is enabled. I always wondered what would happen if one of the kids found the fob in Mom's purse, and tossed it out the window :-). The other end of that spectrum would be a momentary switch connected to a starter relay. That would require the key to be in the ignition and in the "on Position". That also wouldn't address that an actual starter switch cuts power to other things in the car to send full power to the starter, fuel system and ignition. It would work, but I don't think that that is actually the functionality you're looking for.
Or, for around $30.00 from Flea Bay, this might be what you are looking for..
http://www.ebay.ca/itm/Subaru-WRX-SV...vxp=mtr&_uhb=1