Remember, the original purpose of the the Oath Rod (Binder) in the AoL was to Bind criminal channelers, to prevent them from ever committing their crimes again. If the Oath(s) could simply be removed with a weave without the ter'angreal, then Binders would have been useless in the AoL.

Whatever solution Mesaana used has to be something that wouldn't have worked in the AoL. So, it can't be something that she did later, in the privacy of her own room; otherwise, criminals in the AoL could have done the same thing. It has to be something she did at the time she was taking the Oaths, that the 3rd Age channelers weren't aware of, but that AoL channelers would have prevented her from doing had she tried such a trick when being Bound as a criminal.