Kia ora, assuming that the US company doesn’t have an NZ presence at all these are exported services therefore zero rated for GST purposes. Zero rated services are a bit different to exempt services, in that if you are registered for GST you still need to include them on your return, it’s just that the rate of GST they attract is 0% rather than the regular 15%. Taxpayers are able to claim GST inputs (expenses) on costs incurred in connection with zero-rated services.It would be a bit unusual to bill for the expenses through one entity and the services through another. Tricky to articulate why without a big essay, but consider this, I think you are required to register for GST if you have sales in excess of $60k NZD per annum even if they’re all zero-rated. So if you put the contract for the services in your personal name you may need to register for GST in your personal name too (if not already registered).
Also per Andy’s note it is worth giving consideration to your potential liability in the US, if there are any risks with your services (there usually are) it would seem wise to sign the contract with your company so that you have limited liability. Also consider your insurance position, make sure you have appropriate cover for the type of services you are providing.
Hope that’s of some help!