Here are my thoughts:
1) If you are looking for large management consulting firms, then typically all of them sponsor H1B. The only bottleneck is to get through the H1B lottery.
2) And most of the large management consulting firms do a lot of public sector work. Especially in those states where public sector projects are more common e.g. in DC
3) If you are looking for boutique firms that specialize only in public sector work, then the challenge is sponsorship. Small firms with limited budget for this may be willing to sponsor once, but not three times.
Hope this helps!