There are no rules for this; the situation you've described doesn't exist yet; ask your GM
There are (as of now) no rules at all for how the situation you've described would be resolved so ask your GM. In fact, there aren't even rules for repairing airships whatsoever, let alone during a long rest while they are an integrated tool. Any content from Eberron is still playtest as of now so we would have to wait for further details on how those rules work. Finally you've asked us to assume something which very well may not even be possible, a Warforged choosing an airship as an integrated tool.
At the very least the section on "Long Rests" states:
[...] At the end of a long rest, a character regains all lost hit points [...]
Personally I would not call the vehicle "the character", it is an integrated tool, and no part of that description says that you suddenly share hit points.
Finally the intent of the integrated tool is that it does not apply to vehicles. According to Keith Baker's FAQ on Wayfinder's Guide to Eberron:
An envoy warforged has an integrated tool that’s part of its body, and gains expertise with that tool. In the PHB, tool proficiencies provided by backgrounds include vehicles. So can I have a warforged with a built-in wagon?
All things require the approval of the DM. If your DM WANTS to let you have a warforged with a built in boat, that’s great. But the design intent is that this applies to the specifically identified tools listed on page 154 of the Player’s Handbook. This does not include mounts and vehicles, which are detailed on the following page (“vehicles” are on 154, but not detailed). So again, if you and your DM agree, go for it. But it is not the design intent.