I think to find the goverment class and the fundaments of this constitution we have to take on consideation some factors (this is only my oppinion, I cannot tell you about serious studies about the subject):
- How did the megacorp leave the planet?
If the megacorp just left the planet, either whith some time of planning or without it (more probable, but not exclusive, on the rebellion millieu, with the vilani megacorps pull out), I think the most probable goverment to emerge is a tecnocracy, as the worst problem the planet whould have is the shortage of technicians and administrators (probably most of them were corporate men who leaved wih the megacorp).
If the megacorp just vanished or lost contact (reasons to anyone's guess), I think the most probable goverment to emerge is an oligarchy, mostly formed by the former corporate men, who are probably the best administrators or technicians and in a power position to begin with. This can take the form of a tecnocracy, a charismathic oligarchy or a self perpetuating one, or even a boureaucracy.
If the people fought their freedom from a too controlling megacorp, the situation is more open. A democracy is a good possibility, as people whold be hungry of freedom, and it's nearly sure a quind of anti-trust law, to avoid falling again in economic slavery. There can also emerge a leader who proclamates himself 'dictator' (either king, president for live, consul or whatever he likes to be called), that whould probably be charismathic, but may turn into a uncharismathic dictatorship sooner or later. In any case, it's probable (IMO) that the technicians and administrators left on planet whould have a starting advantage. Also one cannot dicard the fighting degenerated onto an anaarchy, but in this case we have not to worry about a constitution.
In this last scenario (independece won by fighting), it's probable the megacorp tries to enforce some kind of embargo or blockade to the planet, further aggravating its post war situation. This may lead to less freedom on the planet, based on the crisis situation. They may even want it to degenerate on anarchy so that the people whould ask to be absorbed again by the megacorp.
- The cultural background and lobbies present on the planet
Surely once the megacorp leaves it leaves behind a culture, and various groups whould take positions to deffend their interests. This may lead to one (or several allied) lobby outmaneovering the rest (or having a good lead start) and making a tailored constitution, a situation where the consitution doesn't really please anyone but at least everybody sees no one else to have won or to a consensus one, depending probably on the animosity among those groups and the relative power.
- The resources this planet has
A planet that is a trade hub or who rellies on trade is more likely to become an open society than an authathic one.
- The situation in the space zone the planet is on
In a millieu as the Hard Times, it's more probable some powerful person/group to take absolut power 'for the duration of the crisis'. This may be the military, some lobby ot a very powerful individual, or even outsiders (vikings wanting to settle and promising deffense in exchange, some mercenary group seeing an oportunity, etc)
- And probably a miriad of more things I don't think about by now