A thought experiment but 370kW cooling is equivalent to heat transfer of 370 electric kettles working at 1000W to heat up 1 litre of water (1kg) each by 100 degrees C in ~6 minutes. In an hour that's 3700kg liquid heated to 100C! Half that power and it's 1850kg at 100C.
Even if we assume that heat would be dumped into the fuel going to the engine first how much can you heat up the fuel and for how long before it becomes problematic?
I think 1MW is the peak capacity that will be used at extremely short periods. It just doesn't seem very viable otherwise. Definitely it will require a very comprehensive automated thermal management system.
Far less than you imagine. It's mostly centered around ensuring that heat is properly dissipated so as to reduce material expenditures i.e. make things cheaper. Cooling land structures is trivial. The only question is how cheaply it needs to be done. It's an economic issue.
Put the same structures in the air which is the second best common thermal insulator after vacuum and the problem becomes technological. Require that the structure has low IR signature and you have a technological problem that will consume millions before it is even properly understood.
Have you seen it in a thermal imager after that period? Batteries and electric engines are also far less prone to deformation and explosion than aviation engines and fuel.
Tank (combustion) engines running at 1,5MW approach a technological barrier but that barrier exists only because it's hard to put more cooling without exposing it to enemy fire which would then cause mobility kill. IR signature is also a factor now.
Think in terms of thermodynamics. Aircraft operate in an environment that acts as an insulator much like space systems. You can't approach that as anything that has a conductive environment.
Still much easier than managing heat in space lol. It's when you go supercruise (especially > Ma 1.5) that heat dissipation through atmosperic cooling (i.e. ram air, engine bypass, surface heat exchanger) becomes difficult and using fuel as heat sink becomes critical. All higher Ma supercruise aircraft has faced this challenge from the Concorde to the SR71.
You're right that having 1MW power generation ability (and we don't know if it does) doesn't mean it can and would sustain 1MW output, I think it certainly would not, just like that 1kw power supply in my PC may never reach its rated power, but it still gives the headroom to run more powerful hardware running at a few hundred watts, and can cover the occasional peak when needed.
Even half of 1MW still mean a gigantic leap from 5th gen fighters. To take that kind of heat away from your electronic components is not difficult with the appropriate space for cooling component (essentially more fancy car radiators), but to get rid of that heat from your plane entirely could be tricky if 1) you need to do this for a prolonged period of time, and 2) you need to do this while supercruising at Ma 2. In that case you better have very large fuel capacity, not because of the energy consumption (still tiny compared to thrust), but because you'll need the cooling capacity offered by tons of fuel. Good thing jet fuel can take a lot of heat with the right tank design, they can run at above 100C.
Some additional thermal management tricks are also available, for example you can trade speed with more atmospheric cooling, you can "supercharge" your cooling ability with expandable heat sinks (like a tank of LNG), and you can "precharge" your cooling capacity by refrigerating the fuel during "less hot" stages of the flight, or just getting precooled fuel from a tanker.
Here is a fun and in-depth read on this exact topic from Beihang Univeristy:
Also from Beihang, a patented precharge cooling system designed for China's "new generation high performance supersonic aircraft":
(in Chinese, use your preferred translator if needed)
