Energy is conserved in Galilean relativity. The thing you're trying to say is that it's not invariant across reference frames.

The answer linked above actually takes advantage of the fact that energy is not the same in different reference frames in order to make the argument work.

I think you are overthinking the heat thing. If you have a train car full of hot water and you slow the train down (extracting kinetic energy from it) until it stops, the water in the train car does not change temperature at all, other than a bit of sloshing around and loss of heat to the surroundings.