I think the USSR was specifically focused upon the US & Europe and after 1968, China. Anything else like the South Pacific was incidental apart from as you have said US and allied communication and Sigint facilities like the Echelon facility at Waihopai Valley and Pine Gap. However they would have had a very strong focus on Hawaii and the US 7th Fleet because of their own Pacific Fleet.
I am in agreement that Australia and NZ are a strategic problem for China if it wishes to pursue military objectives in South East Asia, the Indian Ocean and the South Pacific, be they just projection of strength through port visits etc or through trainexs, navexs &operations in the Indian Ocean and the South Pacific. With regard to the Indian Ocean do not be surprised to see combined Chinese - Pakistani naval exercises in the Indian Ocean for two reasons.
1. There are good relations between Beijing and Pakistan with Pakistan being a major buyer of arms from Beijing.
2. They have a common "foe" in India and any Pakistani - Chinese naval - air ops in the Indian Ocean would cause discomfort to the Indians and to the US.
At present there is nothing to suggest that the Chinese will follow this path. I know that in another thread I have suggested a long term goal of the Chinese will be to be "top dog" but that is a separate issue looking at things in a global context rather than a regional context. The article you cite
China’s involvement in Fiji and Australia and New Zealand’s position | East Asia Forum supports my view that at present there are no signs of Chinese plans to obtain any strategic dominance in the South Pacific. However just because there are no sign it does not necessarily mean that such plans do not exist. The article cited states that the hard line approach taken by NZ and Australia towards Fiji, has contributed to increased Chinese economic influence in Fiji. I note that the article states that the Chinese are keeping Fiji at arms length with quite hard business clauses in the contracts.
The NZDF let alone the RNZN would never agree to converting the two current ANZAC hulls to non combatant role. It also would go against the DWP. The costs alone would not make it feasible. However during the cold war the Soviets used FVs to trail US and allied naval task forces for SIGINT/ENLINT purposes. There is no reason why a civilian designed and spec hulls could not be built and utilised for this purpose. I think that they would have to sail under the white ensign and with naval crews maybe with light armament for self defence, eg., 25mm Bushmaster, .50 cal LAWS, etc. It could be a joint RAN / RNZN op IMHO it would actually work better as a RAN / RNZN op. Also maybe possibility of operating future maritime UAV of it as well.