Force on a relativistic laser source

To someone riding alongside with the laser source (or in the same system of reference, in relativistic terms) the laser is behaving like it always did, with no increase/decrease in power output.

But the scenario of an observer experiencing a light source moving at relativistic speeds have a name - Doppler beaming, also known as relativistic beaming. Quoting Wikipedia, it's

[...] the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of light.

So Doppler-related frequency shift would cause an observer that sees the laser source approach at .5c to experience a higher energy output due to blue-shifting, while observers seeing the laser move away would experience the exact opposite due to red-shifting:

_{Relativistic Doppler effect. A source of light waves moving to the right, relative to observers, with velocity 0.7c. The frequency is higher for observers on the right, and lower for observers on the left. Source}

The laser source consists of a laser gun and a fuel tank.

Force on laser source = force on laser gun + force on fuel tank

Force on fuel tank:
change of momentum of fuel tank over time (depends on velocity)

Force on gun: $F`=F$ (does not depend on velocity)

Force on laser source: $$F'= F + changeOfMomentumOfFuelTankOverTime$$

The force on fuel tank is kind of complicated thing - maybe we are not interested about the force on fuel tank now?

Hmm ... I guess the point of this post is that: Momentum change of laser beam = momentum change of the fuel tank + momentum change of the laser gun.

The point of dividing the force into the force on the fuel tank and the force on the laser gun is that the force on the fuel tank is special. The fuel tank does not feel the force when the fuel with the momentum leaves the fuel tank.