Is it possible to replace the cooling system (radiator + cooler) of the processor with something else, say a copper plate or something else, small in size with a thickness of <200mm
It depends on the processor (i.e., the maximum power dissipated), the plate area, its thickness and surface conditions — natural or artificial convection.
And then it is possible to calculate the temperature.
If necessary, I will supplement the answer with the processor temperature estimation formulas for such a system.
But for most conventional processors, it seems to me, a small plate for the heat sink will not be enough - it’s not for nothing that honey radiators with artificial convection.
Depending on the conditions, you can make a copper heat sink and a radiator with a supercharger somewhere far away, where there is a place for them. It is possible to use a heat pipe instead of a copper heat sink. Or even complete with Peltier elements (cold side to the processor). You can zabubenit water cooling, moving the pump and the heat exchanger somewhere else.
But some kind of heat exchanger is still needed - the heat flows there are quite large. Or use processors with a lower thermal capacity - from netbooks, for example. There will be easier cooling.