The same article says that the Marines will inhabit "three islets of Yokosuka." I only know of one island near the base, Sarushima (Monkey Island); a CNN Travel article about it is here. There are two artificial islands on the Chiba side of the bay that were built in the 1800s for protecting the entrance to Tokyo Bay, and that may be what the article is referring to.
Also, the "airdrome" would have been on Natsushima (Summer Island), which is technically an island, though it is connected to the mainland by short causeways. Natsushima is just north of the Yokosuka naval base and east of Oppama, though I don't remember an airfield ever being there; the page here says it was used until the 1950s. The airfield seems to have had its own patch, however, here.
Sounds like “Big Island” is the Japanese equivalent of “Deer Creek” in the United States.