Random data point, cos I'm a Jewish IP geek: in the UK, the hechsher, the mark which indicates food is kosher, does have the status of a trademark. More than that, it has the special status of a mark of quality, with the appropriate legal protections which don't apply to more run-of-the-mill trademarks. I'm not sure how I feel about this; to an extent, the civil legislature is basically acting to enforce an aspect of Jewish law, namely kashrut. And it's not theoretical, there's case law on this issue.

As for the agunah (unequal divorce) issue, I would be very concerned about the secular state getting involved in that. There's something to be said for the court enforcing a pre-nup, but that's at the level of the state enforcing contracts, rather than getting involved in religious issues directly.

I'm not American so I don't have a knee-jerk reaction about separation of religion from government. But even so, I really don't think it would benefit the Jewish community as a whole to have this kind of thing enshrined in civil law.