I agree with Chris that the categorization you did actually includes different type of companies; while NERA and Frontier Economics do economic consulting (more Microeconomic analysis, Competition policy and Econometrics), Strategy&, Accenture Strategy and PWC do strategy consulting (more similar to MBB, often based more on implementation though). A first decision therefore should be on which kind of consulting you would like to do, that is, whether economic or strategy consulting – they are both great path, but pretty different, and in economic consulting you are likely to find a glass ceiling without a PhD.
Once you have decided which path to take, in case you decide to go for strategy consulting, you would then have to choose whether to go for more niche firms specialized in public sector, or more prestigious, where you may never specialize at all in public sector. One way to meet both the criteria would be to change geography and focus on the Middle East: there, a good amount of the MBB projects are related to public sector as well.
Finally, you should consider in the choice that your preferences may change in the future, and you may decide to switch to something different from the public sector. In this case, MBB would definitely be the option that could keep more doors open.