Answer: they probably can't, and they shouldn't even try. Culture wars are immense distractions. Important issues come to be regarded, not as issues in their own rights, but as battlefields in a larger conflict.
In the case of Catholics in Ireland, John Waters has argued very powerfully that getting locked into a siege mentality will almost inevitably lead to defeat after defeat:
I have often observed that the pattern of public discourse in what we call ‘modern’ society is that, once a controversial issue – one, I mean, running in the face of ‘traditional’ thinking – has been launched into the public discourse, the defeat of what is called ‘traditionalism’ is all but inevitable.
This applies to virtually all the fundamental questions about the public standing and influence of Christian thinking across a wide range of issues, from gay marriage and adoption to the place to be accorded to faith in education. On all of these questions, the debate is constructed to ensure that, to borrow a famous phrase, the ‘modernisers’ need to be lucky just once, while ‘traditionalists’ need to be lucky all the time. The persistent reiteration of the issue in question, and the gradual wearing-down of opposition, ensures that the ‘debate’ can ultimately have but one outcome. And this sense of inevitability serves to intensify the opposition of ‘traditionalists’, but in a manner suggesting that they are, like Canute, merely admonishing the tide of what is termed human progress.
Similarly, Peter Hitchens in this article argues that social conservative should not get lured into the debate about gay marriage, which he sees as a distraction from the much more important matter of saving traditional marriage itself. (I sympathise with his argument, but I think that the redefinition of marriage has to have a knock-on effect on our attitudes to traditional marriage as well as supposed homosexual marriage.)
In Ireland, culture war thinking has led an opposition to abortion or divorce to be mixed up with Irish nationalism, opposition to the European Union, and other issues. Now, I am an Irish nationalist, and I am also instinctively hostile to supra-national institutions like the European Union. But I recognize the folly of running all these questions together as part of a "culture war".
The difference between conservatism and Christianity is that there is a definite body of doctrine which says what Christianity is and what it is not. In Catholicism, especially, this is the case.
But if Christians or Catholics do want to fight under the banner of conservatism (whatever that might mean), it seems important to me that they remember they are conservative because they are Christian, and not Christian because they are conservative.