I think you have three options. Either fly to Innamincka and stay at the pub or the caravans behind the general store, and get them to organise your road transport and dinner, or prevail on Nappa Merrie to look after you, or bring all your gear and camp on the airstrip by the waterhole if Nappa Merrie will let you.
Last I heard from Innamincka two weeks ago the road from Innamincka to the tree is rooted in the early stage, so if you want to do it from Innamincka you are going to have a rough night time drive back to the pub (45k's?). I have no information on the strips except that I think Innamincka had a windsock last time I was there. There are old photos of planes parked outside the pub, but I wouldn't try it these days - too many tourists.
In the past, the Nappa Merrie people have not been either helpful or too bright. They closed the ford, thus adding 28K's to the trip via the Burke and Wills Bridge, because they got sick and tired of pulling the odd 4WD out of the ford. We would all have been far better served if they had just put up a sign announcing the price for a tow was $200.
I think you should try Nappa Merrie, but don't be surprised if they tell you to %^ck Off. They have yet to understand that tourists are a great cash crop.
Anyway, I hope you get to the tree. Its a great spot, as are the resting places of Burke and Wills by the great Cullyamurra waterhole .