So the final momentum of the emitted photons is also 2mv
The final momentum of the each emitted photon is ±mv, for a
total of zero.
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Also, a box containing two counter-propagating photons
has mass, even though either of the photons separately
is massless.This is in accordance with the formula
E^2 - p^2 c^2 = m^2 c^4
which is incomparably more useful than E = m c^2.
Also easier to explain, easier to teach, easier to learn.
Hint: It's the gorm of the 4-momentum.
Bottom line: The mass of the spaceship does not change
and cannot change unless you throw something overboard.
Furthermore, even if we could magically reduce the
mass, this would not cause the spaceship to accelerate.
This should be obvious from the fact that it cannot
accelerate in the frame where it is at rest. Therefore
it cannot accelerate in any other inertial frame.
Also, if you are going to throw something overboard,
you might as well shoot the photons out the back and
use that as thrust.