If it were me, I'd be fairly happy with the records which indicate (independently) that Thomas (Snr) LAWLESS and Catherine, his wife, (obit. 1922) arrived in 1884. [That's "two outta three", isn't it ?
]
Just to explain (briefly) about NZ immigration, 1880's.
Your people appear to have arrived here at a time when the immigration scheme earlier in place (Vogel scheme), had broken down (due to the cost of subsidising passage fares for immigrants).
In 1882 the Government of the day re-instated the funding of the scheme for supposedly another 3 years. However, by
1884, most of the "immigrants" were of the "nominated" variety (rather than being of the "subsidised" kind.) And this factor ("nominated"), would explain why there probably isn't a passenger list record - given that most lists that exist, relate to those who arrived here having been wholly or partially subsidised under a government or other, immigration scheme.
* Do you know if the LAWLESS family had relatives who may have arrived in New Zealand ahead of them ??
Of course it needn't have necessarily been a relative who nominated someone - friends ... and even prospective employers could do so. [For some years there are "Nomination records" held by Archives New Zealand - haven't checked for the 1880's. ]
Thomas LAWLESS (bc 1840) born Co. Meath, Ireland, appears from the 1871 and 1881 UK census returns, to have operated as a "beer and wine retailer" and (1881) "a wholesale and retail beer* dealer". [One transcription has "bar" dealer ?
... think it should be "beer: though.]
There were servants in his household too. That suggests he probably had a few "spondoolicks" to his name and could afford to transport his family to the other side of the world ?? [He may have arrived here simply as a "fare- paying" passenger ... and there's probably no passenger list for that mode of arrival, either. ]
~ Lu