Sometimes it seems that it just doesn't want to give you the chain extension.
After some investigation, I think the easiest way to get the chain off the green airship revolves around not destroying all of its parts. Each of the two wings on the airship can be destroyed individually for 1000 points (not counting the various guns on them), and the core can be destroyed for 500 with bomb, or 5000 with shot/aura. What's important is that the core can actually be destroyed twice, and the second time is what awards the chain extension.
If you destroy both wings and the core, the ship self-destructs right afterwards, so you'll most likely miss out on the second destruction bonus and the chain extension. If you leave one (or both) of the wings still alive, however, the ship will stay alive even after destroying the core the first time, so you can finish it off the second time yourself and get the chain extension.
Bearing this in mind, the safest way of getting the chain extension would be to throw the bomb so that it damages just the left side of the ship, while ignoring the right wing. It's still possible to get the right wing and the chain extension, but you need to make sure you've damaged the core enough before destroying the wings so you can finish it off before it self-destructs.
