Best way to send Records?

Records got forgotten years ago when NZPost first did a major change of all their costs. At that time it worked out cheaper to send a record from here to Scotland than to your neighbour. They fixed that (kind of) after some hard questions from LP sellers. These days it still isn't cheap. Most sellers have costs around $9 to $15. What you can do is have a look at current record sellers and if you find one with a low cost, ask them who they use.  If you really want a company to pack it for you that's going to be very expensive. There are only a few niche companies offering it that I know of. (EG https://www.packsend.co.nz) At a complete guess I would say that's going to be $20 to $30.

You mention you want cheap AND reliable. Those two things together seldom exist in my experience.

Buy some record mailers (cardboard boxes specifically for LPs) from your local record store. Don't pay more than $3 each. Double wrap each LP in bubble wrap. Each box (which should hold at least 5 LPs) will cost around $12 to send through NZ Post. Sadly, Trade Me is now charging a success fee on shipping costs (which will eat into any profit you make) and, at the same time, NZ Post has increased the cost of sending small packages. You might want to try DISCOGS as an alternative to Trade Me as they're not as greedy. As for choosing a courier, my experience is to avoid ARAMEX at all costs. Good luck!