Hello Andrew, and welcome to BYC!
Wow! That is one
small island, can you get water Taxi services out there or maybe a supply barge? Drop ship to Denman and slowly bring it over from there? Our friend on Prevost gets a supply barge twice a year for things like fuel, feed, and for shipping her livestock, as well as any large items.
What do you have for predators on island? Especially thinking of birds of prey, raccoons, and mink etc? That would all effect your keeping arrangement, and Coop set up. For lumber... you could always get things precut smaller for flying in, depending on the size/frequency of the helicopter. Get lots of 4’ long pieces and sister them up to build a reasonable sized little coop? I’m also thinking about your wind exposure and maybe anchoring it down firmly.
Greetings from Saturna Island, just a little South

Kristen (and I whine about B.C. Ferries.

)