I daresay earlier Jan southbound is a better option than later in Jan. The end of the summer school vacation break traditionally means heavy loads southbound from mod to late Jan as a general rule.
Apart from getting ZED's or ID's to cover both carriers, other options are Air Pacific via NAN (probably going to be busy though - heavy VFR traffic and school vacation traffic NAN-AKL), through Tahiti with TN, through BUE with AR or through SCL with LA.