The generator is powered from a heat engine. Thus, 1 MW worth of sustained electricity being generated also means on the order of ~1 MW of sustained heat being dumped too.
I'm pretty sure the distinction you made between heat engine and battery is incorrect.
If you're talking about transforming the mechanical energy at the accessory gearbox of a turbofan into electricity, the efficiency can be fairly high at >90%, heat dissipation for this part would be fairly manageable even at 1MW (requires about 100kw of cooling capacity). Even the cooling system from a regular car would be able to handle that.
If you're talking about the jet engine generating an extra 1MW of power from fuel, and the cooling implication for that......well that's just a small part of what the WS-15 typically generates (most of it becomes thrust), the cooling needs for a turbofan engine is an already solved problem.
If you're talking about cooling for the electronic components (e.g. radar) that eventually used up that 1MW, yes you need increased cooling when you fire up more power hungry electronic devices. But this has nothing to do with whether that power came from a battery, from utility poles, or from a so called heat engine. The source of the electricity makes zero difference here.
Cooling at the component end is also not a difficult problem to solve as long as you have space (a big fuselage). Just like tphuang mentioned, modern cars (regardless of EV or ICE) can easily achieve hundreds of kw in cooling capacity, and 1MW is just 1000kw. 6th gen designers may try to minimize their system size/weight while achieving the needed cooling capacity, but this is just a matter of how compact they could make it, not a question of whether it is achievable.
An advanced NEV tech tree would also help in minimizing efficiency loss (waste heat) in the storage, transmission and distribution of this 1MW.