Just reporting back incase any other Elite Partners have a similar query in the future.

I was not able to access the Gold lounge and was pointed towards the SilverKris lounge - but obviously would not be able to guest my partner in.

I believe AirNZ have changed their partner lounge website as per TK's post - they now state that Elite Partners can access + guest into the SATS lounge when flying either AirNZ or NZ ticketed Singapore Airlines (I swear previously it just said only on flights operated by airNZ).

Went to SATS lounge, spoke to the attendant who wasn't sure so asked their supervisor, who then confirmed we were allowed in. This included the "Executive Lounge" which was just a quiet room in the lounge. The SATS lounge was relatively nice and had some good showers, which was much needed and welcomed!

TLDR; AirNZ's website now accurately reflects what Elite Partners can access in Singapore - SilverKris (with no guest) and SATS lounge (1 guest)