I dont have mach audio or the auto climate, but unless it has any preamp outs on the back its not going to make a difference. the wires you got with your kit were meant to hook up to the internal amp in an aftermarket receiver, not to a separate amp, most of which hook up through RCA (the wire you have are just speaker wire). If the mach system has a preamp output then you should be able to connect to the amp with a long RCA cable and the run your speakers to the amp like normal. If not, then you can you could buy a cheap hu with preamp to mount in your ashtray or glove compartment. either way, you should be looking to return or resell that schoshe kit, because i dont think you have a use for it with your setup. i remember seeing some sort of new device mentioned here that is designed to help people with auto climate improve their audio systems, i think it was a bit pricey though.
also, im wondering why you are only looking to amp your front channels if you have a 4 channell amp?
I just noticed that you called it a factory amp bypass kit, is this different from the rediculously expensive plastic oval with a hole in it? if so, please disregard what you just read, because i have no idea what kit you are talking about.