Stop thinking of them as dogs. To her and the ex they are their children.

They are a part of the life she had before she met you.

Children of divorce end up with 2 families. Why should this be any different?

Would it be possible for her to have the dogs for a weekend every month? She could be with her babies, they could get to know you and the ex would have minimal contact.

Or, you can go with her next time and take the dogs for a walk or whatever.

As for your girlfriend, her sharing of her journal with you is a huge sign that she feels safe with you. Don't discount that.

If you both want this relationship to progress then you must learn to communicate in a way that doesn't trigger anxiety in either of you. If you can't do that yourself perhaps her therapist can help.

Whatever you do, do not get between her and her children. Anything like "it's me or the dogs" and you will find yourself suddenly single.