Ok - having looked at the map and sat views of this area, it really SHOULD be possible to transfer the land out of airport jurisdiction.
https://maps.google.com/maps?ll=33.9425,-118.408056&q=loc:33.9425,-118.408056&hl=en&t=m&z=14
It’s bounded on two sides by roads, and the other two sides by parking lots. One of those parking lots does back on to airport facilities – lighting – but those facilities are directly backed on by a whole string of businesses on the north side of the facilities. If Google maps is to be believed, those businesses are NOT on airport land. This looks like something that sort-of gradually happened, without a clear intent at some point in the past to make the Proud Bird be on LAX owned property.
However, in my cynical experience, the land ownership change will never happen, because it would require all those people who could take action on this to do something other than say “It’s because of the rules, I can’t change those.”