Lots of things can be very bad for women. Marriage, having children. Having children in or outside marriage. The list is endless. You've mentioned quite a lot of harms, which is great. But individually, these harms can occur from lots of other things we're not banning. There are many parents who have several kids born with disabilities, and the parents aren't related. In those situations, regardless parents always feel blame.
My point, which I'll stop belabouring, is that a precise reason (with numbers) would make a more logical argument and maybe a stronger reason.
While of course we shouldn't shy away from the emotional aspect, we need to realise there are lots of things we allow that are empirically bad for women and children and needs to reduce any ill feelings this might cause.
In my view, one way to do this is to be very clear about why this is banned, with the figures to back it up.
There are I'm sure a few things that increase the risk of disabilities, we're not banning each one. Probably because this has a higher risk.
What people might not realise, or in fact care about, is that cousin marriages are part of people's culture as much as having the freedom to have children outside marriage is a part of ours.
When asking someone to give up a culture that's important to them, I think you do better by telling them why and proving that it's not actually about persecuting a minority culture.
All this will reduce the feelings of resentment and oppression they will likely feel, this won't be the first time England is telling them what to do and maybe even how to feel.
However, I suspect most people have made up their minds one way or another for one reason or another