your running a "bridged" 600Watt into 300? ... the 1200 bit is only a maximum but most manufacturers recommmend you dont go over the RMS for long periods of time.
Does sound like the amp might be overheating. How have you placed it in your motor? if its on carpet its a no-no, that stuff keeps it warmer than it should be and if your running the amp at more than peak output not only arent you giving it enough time to cool but it'll be getting too warm and the cut-off will be kicking in. Same can be said for your headunit though, run it for long periods of time on high volume and that can get warm, Clios aint too good for Head units as they sit in front of the front heater feed and that space gets warm quick too.
If it's not that then the other problem may be wiring, there might not be enough Ampage through the wiring for your amp. Check it out and see what happens.