I would just say you are unable to go.
Then wish them well and get over it. It's all you can do without causing trouble in the family.
You didn't take your children to the last two family weddings, and this one has one of the same issues as one of the others. You say it's a six hour drive to this wedding, and at one of the others you didn't want to drive your children back and forth across London.
So in both cases, the travelling is a good enough reason not to go. It's fair enough, six hours travelling is a long time in the car for children.
You don't have to mention feeling upset that your brother's child is invited. It's quite common to invite babes in arms and leave out older children to save on space and cut costs. Especially if you don't know all of or some of the children very well, and you would have to invite lots of children to appease various friends and relatives. You don't really know what's going on with her fiancés side of the family. Try not to see it as a slight to you and your children.
Just decline the invite, say that you can't arrange the childcare you need and with that being the case the travelling time is too much for you to get back if you are needed. If you and your DH really can't cope for a day without your children then you will be happier not to go.
And don't mention the veil. Your cousin might not remember the conversation you had. If she does, she might not realise the significance you have placed on your DD seeing her wear it now. Your DD might not even want to wear the veil when/if she gets married herself.
It's not important. You have lovely memories of when you wore it. Your cousin deserves to make her own memories with it.
Just out of interest, did you go to the other weddings without your children, or did you miss those as well?