Your example has a tricky issue involving angular momentum (see below), but I can address the spirit of the question using a much simpler example. Let us imagine we have a chamber containing two gases, $A$ and $B$, such that $A$-$A$ interaction terms are equal to both the $B$-$B$ interaction terms and the $A$-$B$ interaction terms. (It doesn't hurt to imagine the molecules as red and blue spheres of the same size and weight.)
Now we imagine swapping some of the $A$ molecules with $B$ molecules in such a way that the $A$ molecules all end up on one side of the chamber and all the $B$ molecules on the other. Due to the assumption above, swapping two particles doesn't change the total energy, but by doing this we have created an ordered system from which work can be extracted. (To extract work from this system you need a piston that's permeable to $A$ molecules but not $B$ molecules, and another piston that's only permeable to $B$ but not $A$. See here for details on how, as well as some other relevant stuff about the relationship between work and entropy.)
Now, since we can extract work from this system, will its weight change due to the $E=mc^2$ relation? Perhaps surprisingly, the answer is no. This is because the relevant $E$ is the internal energy of the system (usually written $U$ in thermodynamics), and that hasn't changed.
The work that can be extracted from a system at a constant temperature is given by $U-TS$. By reordering the atoms we've reduced $S$ but kept $U$ constant. When we extract work from this ordered system, its internal energy $U$ also stays constant, but the energy of the environment reduces. Effectively, we take heat out of the environment and convert it to work. Normally this isn't allowed by the second law, because converting heat into work would cause a reduction in entropy - but through a clever use of semi-permeable pistons we can offset that reduction in entropy by an increase in the entropy of the gas mixture.
The point is that the $S$ term represents the disorder (or, more correctly, it represents the opposite of information - see the paper linked above) and the $U$ term represents the energy. The mass of an object depends only on the $U$ term and not on the $S$ term, so order and mass/energy are actually quite independent things.
The tricky thing with your example is that although you keep the energy constant, changing the velocities in the way you describe adds angular momentum to the system. When momentum is involved, $E=mc^2$ is no longer strictly valid, and you have to use the full energy-momentum relation
E^2 = (pc)^2 + (mc^2)^2,
where $p$ is the relativistic momentum. I do not know how to do this for your example. It may or may not be the case that the effective mass would change. However, if it does change it's because the momentum has changed and not because the system has become more ordered.