It is always possible that a new part is defective. I could tell you stories….
Have you looked at the thermostat? I don't know why I didn't think of this earlier. A bad thermostat can cause all sorts of problems with overheating.
The coolant in the overflow should rise when warmed up. If it did not, then the radiator cap is not lifting as it should. This can be caused by either a too high pressure cap or a leak in pressure somewhere else in the system. The problem with using the higher pressure cap is that if there is a weak spot in the system the fluid will come out there instead of the overflow tank.
I would try to stick with the recommended cap and fix any leaks that you have. There is a radiator hose on both the left and right side of the radiator. There is also probably a small coolant return hose on the thermostat housing.
At this point keep checking that there isn't any antifreeze in the engine oil. A blown head gasket, rarely, can put pressure into the coolant system. Also, when running, pay attention to the exhaust. Just a precaution.