Coral Sun Airways in Tarawa. Recent thread about them and a job advertisement in the Reporting Points section of DG&P.
Air Kiribati in Tarawa too, not sure what they need.
Solomon Airlines take CPL+MECIR freshies for Twotter F/O positions. $12K/yr but just about everything other than food and grog paid for. Not sure what it would really be like for families, but you wouldn't be the only expat one over there.