I am a diocesan priest in the parish of Cyrill and Method in the Archdiocese of Vienna, and a teacher at the How should a rational civil authority regulate marriage, supposing that the state does not accept a particular religious revelation or tradition as definitive, and that the citizens of the state recognize various religious authorities? Since most of us live in civil states like this, the question is not unimportant for us. This is the actual situation.
Two basic possibilities suggest themselves: the state could make the decisions about marriage, or the various religious authorities could make the decisions.
(1) The state could alone make decisions about marriage: e.g., judge what are the conditions intrinsically required to marry validly; establish impediments to marriage; judge whether the requirements of the natural law admit of exceptions, and if so, in what instances exceptions should be made.
(2) The state could leave all decisions about marriage to the various religions.
The first possibility is problematic, since marriage has an intrinsically religious character, and therefore judgment and determination about it should properly belong to religious authority.
The second possibility cannot be consistently upheld, if the state itself is to have any recognition of marriage, and if it is to uphold natural law. For in an instance where the spouses belong to different religions, or after a religious conversion where the second religion has a different view on marriage, one religious authority may maintain that a marriage is valid, while the other religious authority maintains that it is invalid. Moreover, a religion could mandate conditions of marriage contrary to natural law (requiring women to marry even without their consent, allowing divorce and remarriage under any conditions, at the simple request of the spouses, etc.).
It seems therefore necessary to qualify the qualify the second possibility.
(2b) The state could leave decisions about marriage to the various religions, unless the pertinent religions have no position or disagree, or a religion stipulates something about marriage contrary to that which natural law considered in itself requires, in which case the state must decide the case.
Of course, one who is convinced of the truth and reasonableness of his faith will see the ideal as a situation where the state recognizes that revelation as true, and accepts its religious understanding of marriage, but where that is not possible, this second approach seems to be the best.
What are the consequences of this position? Consider the quality of marriage that it be between one man and one woman, and that it be indissoluble. Now, if these qualities belong to natural law, and according to natural law admit of no exception, it seems the state should refuse to recognize the dissolution of a marriage by the Catholic Church (by way of the "Petrine privilige") or through a second marriage in the faith (by way of the "Pauline Privilege"). This consequence doesn't seem very desirable to most Catholics; nor does it seem to be wished for by the 20th-century Papal writings on marriage that address political issues.
If, on the other hand, though the qualities of marital unity and indissolubility belong to natural law, they are the kind of natural law admits of exceptions, then the state can do one of the following: (1) recognize all exceptions that religious authorities recognize; thus the state will accept all divorces accepted by Catholics, protestants, Jews, Muslims, all second marriages recognized by the religions, etc; from a Catholic point of view, this also seems problematic; (2) recognize those exceptions that seem particularly reasonable; this doesn't seem ultimately all that different from the first approach, where the state itself make decisions about marriage; nonetheless, I wonder whether it might not be the most reasonable approach for a civil authority that recognizes natural law but doesn't accept one specific religious tradition as being the recipient of the fullness of truth.
Regardless of the particular answer one might give to the question, "How should the state regulate marriage," I think it's important to reflect on the question, and to have some coherent answer, rather than, with the claim to be arguing from natural law, to push essentially for an assimilation of the state's regulation to that of the Catholic Church–this kind of arguing can give the impression of simple bias and partisanship.
Some version of your final suggestion seems necessary to me. The question then is how a state should determine what are "reasonable" exceptions to the natural law.
My best guess is that a state would concern itself with those aspects of natural law that have most to do with the common good. Because marriage is the foundation of society, stable marriage is something the state should promote; because divorce-on-demand and gay marriage lead inevitably to the widespread instability of marriage, states should oppose them. Since it does not seem likely that the Pope will busy himself with dissolving marriages on every side, allowing something like the Petrine privilege, even from the point of view of an unbelieving state, would not endanger that stability necessary for the health of the state. If something like the Cathar religion came along, for example, the state would have to oppose that.
Although I have not thought it through carefully, I suspect there are many cases in which the state should only enforce morality–natural law–to the degree necessary for the health of the state.