How does conservation of energy explain the Doppler effect?

According to modern physics (although I don't know exactly what the theories that back it up are,) the energy of a system is conserved within a frame of reference, such as in the example of a person on a train throwing a ball. Although the speed of the ball varies for each person's reference frame, the total energy is conserved for each of those frames.
I was searching for the conservation of energy and found a 'confusing' topic related to it.

Conservation of energy and Doppler effect?

Redshifted Photon Energy

The question is, how does conservation of energy explain the Doppler effect?
I get that this question might be (and probably is) a duplicate because I read the questions above and got explanations for each of them. But I felt a bit unsure because of the following scenario:
Imagine a bicycle (probably pedaled by Chuck Norris) moving away from a stationary observer. And let's say that the air around it is very hot and the bicycle starts glowing because of it.
If I understood correctly, then the wavelength of the light seen by the observer and Chuck Norris should vary because of the Doppler effect. However, it seems like the same amount of energy has been added to the bicycle regardless of what reference frame we choose to be in, since the energy in the room is the same. Hence, the energy emitted as light (the wavelength) should also be the same.
I'm quite sure that one of my declarations is wrong, which leads to a wrong result like above. But I'm not sure which one. So the detailed question would be:

1. What wrong assumption did I make in the above example?
2. What would be the explanation of Doppler effect in the sense of conservation of energy?

This is not hard to see using special relativity, but I'll give you an even easier classical example. Say the bicycle and Chuck Norris together have mass $M$ but he is also carrying two basketballs of mass $m$ each. The bicycle is moving away from you at velocity $u$. Now Chuck throws the basketballs in the forward and backward direction at a velocity $v$ which is much bigger than $u$.
You observe the basketball's kinetic energy as $\frac{1}{2}m(v-u)^2$, which is 'red-shifted' compared to the value $\frac{1}{2}mv^2$ in Chuck's frame. But the total energy of the two basketballs is $$\frac{1}{2}m(v-u)^2+\frac{1}{2}m(v+u)^2=2\frac{1}{2}mv^2+\frac{1}{2}2mu^2$$ The first term is the same as in Chuck's frame. The second much smaller term is the extra kinetic energy in the observer frame due to the mass of the two basketballs.
You need this extra term since the kinetic energy of the bicycle system went from $\frac{1}{2}(M+2m)u^2$ to just $\frac{1}{2}Mu^2$