I dont know what traction motors you are planning, but is there a need to preheat them before you push off the line?
If not, why not externally cool them before you launch and then delay liquid N cooling until you are close to their upper spec range? This could mean less coolant tank weight for your build. We ran ice water through the motor and inverter of the Prius BEFORE each run in 2004. The engineers said we were 10% more efficient (in the early part of the run) to leave the line with the motor-generators very cold.
There is very little you can do for the traction issue in a sub-1100 pound car. Your early acceleration is least inhibited by aero drag, and will not be your window of maximum calorie production during the run. With a fixed drive ratio (just guessing you will be around 4:1?), your three phase energy events have more actual time between peak current flow. That final heat build will also be less if you can delay your transition from sine wave to square wave. That was part of the reason I chose the particular speed to transition that Prius, where we had a 250 C limit on certain components (which would commit automatic shutdown of the inverter).
This could mean regulating the cooling in order to keep the operating temp at the highest practical delta for best heat transfer efficiency. Your period of max coolant flow rate may only be about a 40 second (or less) window to bump that record.
As an aside, we found our best speed with that Prius was to attain our peak speed in the middle of the measured mile because the battery and electronics heat was at the point of "diminishing returns". After that point the car began losing speed at a gradually accelerating rate (despite the opposite effect of aero drag). As you can see, the push out method and timing was very important to the result.
We had the IC engine to blame for part of that problem (hybrids steal a lot of power from the engine to try and maintain the battery...net gain at high speed is basically nothing for Bonneville use). You might find a similar approach to your time/speed plan that would allow you to manage cooling capacity while reaching your goal. Picture a speed curve, centered on your timed mile, with the least practical loss during the last half of the mile as you prevent complete thermal forced shutdown.
I am not very well versed with this electric power stuff, but I keep thinking about that best available speed out of 1100 pounds of "stuff" that has ALL its available energy stored before it leaves the line. Your AVERAGE speed for that final measured mile is all that matters. There have been many of us put in some decent records with a car or bike that was "slowing down at the end" where heat build is the final issue.
It doesnt matter, as long as you can work out your plan and manage it. Thanks for joining this forum with your interesting project....the doctors say it is healthy for us old people to put our brains back to work, now and then!