Technically speaking, if the order says that neither party is to "Bring" a third party to the exchange, and you're not WITH the third party, then you're not BRINGING them.
Frankly? If it was written to exclude a specific person, it would have been better just to name that person. A "third party" is also a sibling. So, technically, if you showed up at an exchange with a sibling of the child/ren, or a friend of the children, then you'd be "bringing" a third party to the exchange.
If it were me? I'd continue allowing the new wife to pick up the kids, you keep sending whoever to pick up the kids, and everyone just take a deep breath and dig out the big-kid panties.