But why does the CHT go up? Because the peak cylinder pressures go up. Just like your air compressor in the garage.
This is exactly backwards. Your engine is not an air compressor.
In an air compressor, the piston is adding energy to the gas (compressing it) which heats the gas. In an engine, the piston is extracting energy from the compressed gas which actually cools the gas and lowers the EGT.
You can see this in e.g. CO2 engines which run from compressed gas without combustion. They get (very) cold, not hot. The more power and higher pressures from the engine the colder they get. The cylinders have fins to try to avoid the engine getting too cold, which causes the power to drop off.
In an IC engine the temperature creates the pressure, not the other way around.
The NACA report 754 alludes to this in the section on the effect of timing on valve temperature:
"The rise in valve temperature with greatly retarded spark is probably caused by the higher exhaust-gas temperature resulting from a decreased expansion after combustion"
Earlier combustion = higher pressures but greater expansion after combustion (i.e. piston stroke after combustion) = more energy extracted from the gas = lower EGT but yes, probably higher CHT.
So increased pressure correlates with higher CHT, but it's NOT like an air compressor.