I can't believe I didn't recognise the Heffernan/Plane father-forename swap in the baptismal register! That's what happens when you try and do genealogy sleep deprived!
So, I am going to agree with the thoughts that perhaps Mr. Heffernan is the father of the twins, and that Mary registered them officially with Thomas Plane as the father as he was her lawful husband at the time. The possibility of Edward G. K. also being Heffernan's son is strong, and it would give you a pretty solid explanation as to why only he and Dorothy/Doris accompanied their mother and Heffernan to N.Z. - they were his kids. The elder three (Fred, Charles & Hetta) all belonged to Plane, so they stayed in Aus. (Incidentally, did Edward and Doris keep the name Plane in NZ, or did they use Heffernan?)
Plane seems to be 'officially' shacked up with Harriet Horton by 1908, when their child is born and registered with them as parents, however Harriet's illegitimate child born in 1903 (Arthur Raymond Horton, #2834) later took the name Plane as his own, and on his death certificate Thomas Plane is listed as the father. This seems to me that Thomas and Harriet were together in 1903, so it stands to reason that Mary and Heffernan could be together by 1906.
Now, I am heading Albury way just after Christmas, I can see if I can get a look at the registers of St. Patrick's if you like, and tell you whether there's anything of interest there.
Toban