You say "at most" you can run for 4-5 hours, but how quickly after starting the game are you seeing problems?

Thing is, if it's a PSU problem then I'd expect it to hit pretty soon after and not take 4-5 hours before tripping, assuming you're playing the game pretty steadily over that period of time. Heating OTOH is more nebulous in my experience and ambient temperature and other factors can affect its' onset.

I agree with you that your 600W PSU would seem to be perfectly fine, assuming correct operation, and I agree with the others that getting temp. readings to see how near CPU and GPU thresholds you're at could indicate the problem and prevent you buying an unnecessary PSU.

As for your non-name PSU, you could try running voltage monitoring software if you got something with your motherboard's package. While BIOSes often have a voltage readout in my experience that's pretty useless for this sort of potential problem as by definition you're not driving a gfx card hard when staring at a BIOS TUI.