The network was designed to provide a 180-degree phase difference, NOT constant +90 and -90 degrees required when you ground the middle of the dipole. Floating the dipole with the transformer fixed the problem. The minor common-mode effects of using a physical dipole without a physical transformer should not pose a problem if the network is mounted near the dipole.
It is usually good to employ a balun at the feed point of a dipole when using an unbalanced (coaxial) transmission line. With the network, you would also need a balun between the network and any coax line to the dipole. Mini-Circuits sells baluns and isolation transformers for low power (< 1 watt).