I can see there being a motivation to automate it, if only to make it more pleasant to use. The pump stroke is so long that it might not be too bad to still have the rate of fire of a pump so long as you don't have to wear your arm out like that.
It presents kind of a fun problem because without a good high pressure gas like CO2 or HPA around, traditional pneumatics just wouldn't cut it. I wonder if you could use a much larger solenoid to actuate the priming chamber.
The heat buildup presents an interesting but not altogether unsurmountable problem. Water-cooled barrels anyone? Nothing wrong with a couple 12v fans blowing parallel to the ignition chamber across some heat fins. Or you could install a heat sensor around the breech area that automatically puts the gun in safety when it gets too hot. The C3 is already an electro so that shouldn't be too hard. In fact, I wonder if Tippmann considered putting in a heat-kill ability during its development cycle.
I guess we'll just have to wait until the Brian Cache is discovered and we can build double heat-sinks...