This isn't a street car, is it?
The usual way of dealing with fuel heating is to run the pump flat out all the time and have a pressure relief valve that dumps the excess unused fuel back to the tank possibly through a fuel cooler. Its much more likely to overheat sat almost static in the fuel rails.
Running the pump at full will just cycle the fuel through the fuel rails much faster
This isn't a street car, is it?It is, yes.
Thats the idea. It has less time to heat up. It isn't putting fuel into the rails thats the problem, its it getting heated up whilst its in there, the greater the flow through, the less it will heat, the ride back to the tank and then it sitting in there cooling for a time before going round the loop again. Its the way it's normally done in extreme scenarios like common rail diesels where fuel is pressurised to hundreds of atmospheres for injection. It is bled off through a pressure reducer back to the tank often through a fuel cooler. Just reducing the amount of fuel going into the fuel rail will make it get hotter. In the old days before common rails and circulating the fuel it wasn't unusual for diesels to vapour lock and stop especially after being reduced to idle after an extended period working hard in hot weather. The low fuel flow at idle with the engine still very hot was a killer.
Could you tell me where you live, so I'll know not to go near?
The Fuel Lab pump looks like it accepts a 0 - vehicle voltage, pwm signal. It needs to be between 500Hz and 1500Hz. The default PWM output of the Arduino is 488Hz. In terms of frequency I'd be surprised if you couldn't get away with that. However I wouldn't be surprised if the noise it generates is annoying. In terms of voltage - you're going to need a mosfet. Unfortunately the manufacturer gives no clue regarding the current requirements of this so it's hard to suggest actual components.
Is it at all possible to bump up the frequency of the Arduino?