Just to clean up one item: I did assume in the above that the rest mass of the billiard balls was unchanged in the inelastic collision. But we can drop that assumption and still prove that, if energy is not conserved in the center of mass frame, 3-momentum cannot be conserved in any frame other than the center of mass frame. Suppose the rest mass of the balls after the collision is M (instead of m). Then we have for the 3-momentum before (P0') and after (P1') the collision, in a frame with velocity u relative to the center of mass frame:

P0' = 2mu/sqrt[(1-u^2)(1-v^2)]

P1' = 2Mu/sqrt[(1-u^2)(1-w^2)]

But we can simplify this by writing down the total energy before (E0) and after (E1) the collision, in the center of mass frame:

E0 = 2m/sqrt(1-v^2)

E1 = 2M/sqrt(1-w^2)

So we can see that

P0' = E0 (u/sqrt(1-u^2))

P1' = E1 (u/sqrt(1-u^2))

Hence, if E0 > E1, we must also have P0' > P1. In other words, the only reason we happen to have P0 = P1 in the center of mass frame is that being in that frame is equivalent to having u = 0 in the above formulas.

(It is also straightforward to show that if energy is not conserved in the center of mass frame, it is not conserved in any frame. So the Lorentz invariance of the two conservation laws, energy and momentum, cannot be separated--they are inseparably linked.)