I see it now. That 62C/W is a very odd spec. If it were true, then the 6cm2 of copper is almost totally useless - which I do not believe. It does look like a typo.
You really do not need much to get rid of 300mw. If you had as little as 1cm2 of copper near the device, it would probably be fine, as the copper is not just radiating the heat, but is is also dumping the heat into the bulk of the PCB. I am pretty confident that 2 cm2 would be very safe.
Richard