We can reduce this to 1+1d, the box is just a bar of length 2L in its rest frame. We set c=1.
Box rest frame :
Coordinates at (t,x). Ends of box at $$x = \pm L$$. A light sphere is emitted at t=0 from x=0 and thus has equation $$x^{2}=t^{2}$$. The LHS end is reached when x=-t=-L, ie t = L. The RHS end is reached when x=t=L, ie t=L.
Frame where box is moving to the right with speed v :
Coordinates are (T,X). Relativity says emissions of light sphere at X=0, so equation is $$X^{2} = T^{2}$$. Box moves in positive X direction at speed v so is Lorentz transformed to be of length $$\frac{L}{\gamma}$$. Ends of box are thus at $$X = \pm \frac{L}{\gamma} + vT$$ where - is left hand side, + is right hand side. LHS is reached when $$X = -\frac{L}{\gamma}+vT = -T$$ so $$T = \frac{L}{\gamma(1+v)}$$ and thus $$X = -\frac{L}{\gamma(1+v)}$$. RHS is reached when $$X = +\frac{L}{\gamma}+vT = +T$$ so $$T = \frac{L}{\gamma(1-v)}$$ and so $$X = \frac{L}{\gamma(1-v)}$$.
Summary so far :
So by just doing simple geometry we have in the rest frame the LHS is reached when (t,x) = L(1,-1) and the RHS when (t,x) = L(1,1). In the moving frame we have the LHS at $$(T,X) = \frac{L}{\gamma(1+v)}(1,-1)$$ and the RHS at $$(T,X) = \frac{L}{\gamma(1-v)}(1,1)$$.
We can clear this up a bit by defining the Doppler shift $$D(v) = \sqrt{\frac{1+v}{1-v}}$$ and using $$\gamma^{-1} = \sqrt{1-v}\sqrt{1+v}$$ we get the LHS at $$(T,X) = L\,D(-v)(1,-1)$$ and the RHS at $$(T,X) = L\,D(v)(1,1)$$.
Lorentz Transforms :
To go from (t,x) to (T,X) we have boosted by a speed of -v in the x axis (hence why the box goes to the right in the moving case), thus the transformations are
$$T = \gamma(t+vx)$$ and $$X = \gamma(x+vt)$$
ie what you get on Wiki but with a negative v. So let's apply these to the rest frame coordinates.
LHS :
$$(t,x) = L(1,-1) \to (T,X) = \gamma ( L - vL , -L+vL ) = \gamma L (1-v) (1,-1) = L \, D(-v)(1,-1)$$
RHS :
$$(t,x) = L(1,1) \to (T,X) = \gamma ( L + vL , L+vL ) = \gamma L (1+v) (1,1) = L \, D(v)(1,1)$$
Precisely as we had from doing each frame by simple "When do these two things meet?", the Lorentz transforms indeed give the required answer. And this used the $$x^{2} = t^{2} \to X^{2} = T^{2}$$ property of relativity which maps light spheres to light spheres.
I haven't had to put in any numbers, this is a completely general case. So, what's the contradiction MD? Relativity agrees with itself, it has not been inconsistent as you claimed.
Now how about you answering my question, if you're so damn sure you're right and you've got so much time on your hands and you've known this for years why are you still whining about it on forums?