You are going to kill your diode.
Instead of ruining a rare item, why not just sell it to someone that will make some use of it?
Temperature is
not the only concern when you overdrive a diode. For example, if you take a 5mW CD ROM reader diode, it does not get warm at its operating current (~45mA). Hell, even at 200% this, you won't feel any warmth. This does not mean you can push the diode until it gets hot. Eventually the physical structures of the die will give out before you can measure any change in temperature.
Even with a TEC to regulate the temperature, there is a limit to how much energy the tiny semiconductor can handle. The facets of the die will start to degrade and you will end up with a useless "LED."
The wavelength
will shift toward red as you increase the current even if you keep it cool. The difference may not be immediately noticeable if you have no other "635nm" for comparison.
Why not appreciate the diode for what it is? If your only concern is power, don't buy a rare diode and destroy it trying to squeeze out every last photon. Trust me, it will only end in premature death.
The value of this diode is NOT that it is ~$30. The value is that
the wavelength is 638nm @ 40mW with great beam characteristics.
So what is the point of getting 81mW out of it if it is lasing at 645nm? Why not just buy a 100mW 650nm diode?