I myself can think of an argument in favour of the Burka women which is better than both. It goes like this [1] its prima facie wrong to require people to act in violate their religious beliefs and act in a manner that they consider themselves to be engaging in the equivalent of a pornographic photo shot, in order to get access to public services like roads. [2] requiring women to take the Burka of for a drivers license does this.

Now I think [1] is plausible, I would base it in part however on theological premises about the primacy of conscience, which much western commitment to freedom of religion is historically based on. I doubt I could give a knock down argument from premises all people accept to its truth, but in this it is no different from any argument for any substantial issue of public policy. I suspect however that this theologically based argument would be far more persuasive to most people than either we have discussed so far.

Really? In the example of the driver’s licence photo, the arguments on one side are valid; the argument of the other side amounts to “God said so,” which is in no sense a valid argument.
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2.In response to the Burka issue you write This would be plausible if there was actually an argument for stripping naked as a requirement for getting a license. this begs the question, if the Muslim womens beliefs about Burka’s are correct, the state is demanding the equivalence of her stripping naked to get a license, that would clearly be an unjust law, the only reason you can say this is not occuring is if you assume the Muslim’s beliefs are false, from the outset.

You go on to suggest that the “ the basis for putting a picture on a driver’s licence, the argument for which is way more compelling than anything a woman wearing a burqa for religious reasons can come up with.” this is because “ The premises are pretty straightforward. In some situations authorities need to identify the driver of a vehicle (and for the purposes of this discussion, let’s leave out why they should be allowed to do so, or we’ll have written a book by the time we’re finished); therefore the licence that the driver presents to the authority figure should allow said authority figure to identify the bearer as the owner of the licence. The means of establishing that should, in the interests of efficiency, involve as little time as possible, the minimum of equipment and the minimum of expense. Thus far, no-one’s come up with a system that better meets those criteria than putting the owner’s face on the licence.”

Actually the second premise of this argument is I think false and fairly uncompelling it suggests that the means employed in driver identification should be as cheap and efficent as possible. Thats mistaken, it should employ the cheapest most efficent morally permissible means. One can never engage in an immoral means because its cheap and efficent. Hence this secular argument is not compelling at all.

In light of that, someone who wishes their face not to be on the licence must come up with at least equally compelling and equally practical reasons why not. What compelling, practical reasons are available in this case from appeals to the supernatural?
Well the women would argue from passages in the Koran and Hadith that Allah commands her to wear a burka. This is clearly practical, it after all tells us what to do in practise. Is it compelling, well it might be to those who accept the authority of the Koran and the Hadith. Of course it won’t be compelling to those who do not. But in this its no different from the argument you give, as I noted above the argument you give has a false premise and hence one many people can and should reject. Neither argument appeals to premises accepted by all rational people.

3 You conclude “I return to my original point: the asymmetry here is between arguments that are useful and arguments that aren’t.” The problem is you have not shown that religious arguments are never useful and secular ones are. If by useful you mean arguments that are likely to convince others, then both will appeal to premises that some people accept and others do not. The religious person will appeal to religious claims, the secularist to secular moral principles, neither will appeal to premises accepted by all rational people. Both are as useful then as each other.
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And all point of view ends up appealing to the ‘supernatural’ in the end. Not in all cases, but in some cases, especially on moral issues.