I think actually it's the word 'law' that we are using differently. It sounds as though you are using "law" to refer to the absolute divine law ordained by God (I'll call that Law with a capital L) and using "government" to mean the secular legal structures (law with a little l) developed by people. 'Morality' = 'Law' in your construct, and ideal law should be an approximation of Law. (This is my interpretation of your post - correct me if I'm wrong.)
My view is a little different: I see Law as something that is not absolute, but will vary between cultures, religions, etc. The Law for the Hindu is not the Law of the Christian, and so on. It doesn't need a religious base, for example a vegan will abide by a moral structure (a secular 'Law') that has been arrived at philosophically, perhaps without reference to a higher power. Individuals may abide by a different Law, depending on culture and upbringing.
On the other hand, law (small l) is the structure that a society creates in order to achieve the secular goals that I laid out. Individuals within a society may follow different Laws, but they are all subject to the same law to allow the society to function. It's this law that I believe should not have a basis solely in morality, because members of the society may not share a common moral structure. Small l law should impinge on personal freedom (and personal adherence to Law) as little as possible.
I do agree with you on the idea that a shared morality is part of the glue that binds a culture, and a society with shared cultural values is a stronger society. However, I don't think that law is the correct mechanism for producing this cultural sharing. (In the real world it's not as clear-cut as I describe - interactions between 'society' and 'culture' tend to dilute the idealized version of law that I describe, but that may not always be a bad thing in a pragmatic sense, although I might dislike it in an academic sense.)
Anyway, thanks for your substantive response - I've enjoyed thinking about the topic even though there probably won't be many minds changed on this thread...