I wouldn't list your entire CAF, but they would expect most parents to be looking at 3 schools, so no harm in listing another private and your preferred state.
I've just read over your response to DG:
We have applied to NLCS and Habs for the following reasons:
1. Oxbridge transfer from these two schools seem to be much higher than DAO.
2. overall results and position in the league tables suggests that the above two are better placed than DAO.
If this is what you are making judgements upon, then you need to ask Habs and NLCS closely about what proportion of their 6th form are new joiners, as well as what the criteria is for staying on after GCSE and also after the 1st year of 6th form. I often find that schools can make a significant jump up the league tables by, say, not letting anyone who got a C or worse at AS sit the A level. Now, I don't know the schools that well, and I am sure that there will be other differences, but essentially league tables results can often be improved by restricting the options available to pupils. No idea if that is at play here, but it is worth enquiring about - eg how many pupils do not continue to the 2nd year of A levels.