Wow! I went and checked this - somewhat disbelieving to be honest - and there it was in the list of prohibited goods: artwork and sculpture, a few lines down from guns, ammunition and bullets 
I know we all believe that art is powerful, but this is ridiculous.
I went on Google and had a rummage, like you do, and there’s DHL, FedEx and parcelmonkey.com
I asked for an estimate from Parcelmonkey based on a guess at what the painting might weigh - based on my own shipped stuff - and the price was fairly eye-watering - over $48 - which is more than my work often sells for so it all depends on your pricing I guess and the enthusiasm of the buyer.
The only other way I have shipped (to the US from here, the UK) when the cost would be too much via Royal Mail is when the customer was a commercial gallery. He paid the shipping by giving me the details of his own FedEx account which I used at this end.
If you have that sort of customer this might be a solution.
I notice USPS have quite a few countries on their banned list at the moment due to Covid-19 and I vaguely remember reading that NZ has all but closed its borders so it is possible shipping might be hard at present using anyone. Good luck 