The most rational way to get to NAN is via LAX and Fiji Airways. They offer reasonable fares and good service, and a bit of island charm to boot. Not a DL partner, but you can credit to or redeem miles from Alaska if that works for your needs.

The most SkyTeam way to get there is on Korean Air via ICN. We did this after our wedding for our honeymoon in 2015 and it was a lovely adventure, but probably ill advised in the final reckoning - HNL-ICN-NAN was a loooong slog, and we were pretty cranky by the time we got to Fiji. (The free honeymoon cake KE provided us helped matters, though, as did the upper deck on a 744 for the first segment.)

If you’re dead set on sticking with DL partners and metal, the way I’d do it is via SYD/MEL/BNE and Virgin Australia. The 737 flight to Nadi is pretty grim - their premium economy product is lame and economy is lamer - but it gets you there in only a few hours of backtracking, and you can add a vacation down under to extend your beach fun. We’ve transited all three; of the airports, Brisbane is by far the worst, like an Aussie MCO. Avoid if possible, although it tends to have the best award availability, so... 🤷

Other Asian hubs don’t make a heck of a lot of sense, so I’d skip routes involving Singapore/Hong Kong/Denpasar unless you’re desperate for every mile.