I really don't know what to tell you, to be honest. You don't need her approval for what you and the kids do during their time with you, and it's obviously not in their best interests not to meet their sibling. The problem is that she's probably not going to know, or rather accept this until someone with authority tells her how it is, so bottom line is that you really do need to commence mediation to start trying to get this sorted, and I would be doing it sooner rather than later.
Your plan for October holidays is fair, but I wouldn't wait until then to start mediation. It can take up to six or eight weeks to actually get mediation booked and sorted, sometimes longer - we once commenced in April and it didn't progress until August, at which time we were told the matter wasn't suitable for mediation because of how much difficulty they were having trying to get the mother's agreement to attend.
So don't wait. Just start now, and advise that in the event agreement can't be reached, you will be pursuing the matter further through the Family Court.