Delta Has Access to More Air Tahiti Nui Space than You Find on Expert Flyer

It seems that experflyer.com is not accurately displaying Air Tahiti Nui award space that Delta agents can book. Multiple Delta representatives confirmed with me over the phone that four seats in Business Class are available on seven different Air Tahiti Nui flights, all of which show zero award seats in Business Class available on expertflyer.com.
What Air Tahiti Nui Is and Why it Matters
Air Tahiti Nui, headquartered in Papeete, Tahiti, is one of seven airlines that flies into Fa'a'ā International Airport in Tahiti.

A System to Get to Europe with Delta SkyMiles

I use three steps to get to my European destination with Delta SkyMiles. For each direction:

Find transatlantic space.
Find domestic space within the US on Delta or Alaska from my home airport to the international gateway.
Find intra-European space to my destination.

Those steps are roughly in order of difficulty--though many times finding domestic space on Delta is impossible at the low-miles price.