When you coast with engaged clutch, you will engine brake

True

   Energy doesn't go to the engine for free.
   Adding the fuel to counter the engine braking is less
   efficient than letting the engine run idle while coasting
   due to losses in the power-train

It is a bit hard to understand where you're trying to go here but assuming that you state it takes fuel to 'counter the engine braking' you're wrong. Engine braking is just that, using the engine as a brake. The engine is used as a pump, air (and only air, no fuel is added) is compressed, the air heats up, this heat is dissipated by the cooling system. That is where the energy actually goes, as heat. Losses in the power train only add to the braking effect.

This is also why it is a bad idea to use engine braking with a 2-stroke petrol engine: 2-strokes rely on oil mixed in with fuel for lubrication. When using the engine as a brake no - or hardly any - fuel is added to the incoming air while maintaining high engine speeds, starving the engine of lubrication.