Nice to see you here too, Venerable. Those are good questions. We discussed this briefly before. Here are my thoughts.
SN48.40 is problematic for both interpretations of the jhānas, actually, because the “pleasure faculty” (sukha-indriya) is said to be abandoned already in the third jhāna, while sukha is normally said to be abandoned in the fourth. This is awkward no matter how we interpret the jhānas.
If we interpret the sukha of the third jhāna to be experienced “with the body”, then why does SN48.40 say the faculty of sukha (which SN48.36–39 define as bodily) has ceased in the third jhāna? If anything, this then actually confirms that the sukha of the third jhāna is not bodily, because it says there’s no more bodily sukha in the third jhāna. So the sukha of the third jhana must then be mental. 
But then the sukha of the second jhāna apparently is bodily? How does that make sense? Does the meaning of sukha shift between the jhānas? In the 1st and 2nd it is bodily but in the 3rd it is not? That seems strange. 
In AN5.176 all pleasure (sukha) and pain connected to the kāma are abandoned in the first jhāna already, so this includes bodily pleasure. This apparently contradicts SN48.40 as well.
It’s likewise strange why the faculty of domanassa (“sadness”) is said to be abandoned in the second jhana in SN48.40. This also contradicts AN5.176 which implies all domanassa (of the skillful and unskillful and of the sensual) is already abandoned in the first jhāna.
Now, the answer may be authenticity, as you suggest when you say “at least one must be wrong”.
But if we assume the text is authentic, then I would reply that the bodily sukha is already abandoned in the first jhana, but in SN48.40 it’s linked to the third jhāna exactly to make the point that the sukha that still exists in the third jhāna is NOT bodily. In other words, bodily sukha was already gone at first jhana, but to emphasize that sukha in the third jhana is mental, it’s stated specifically before the third jhana that bodily sukha has ceased there.
Because technically speaking the text actually does not say that bodily sukha still exists in the earlier jhanas (although we may imply that). It just says that it has ceased in the third. That’s being a bit pedantic, but it does seem to me the best way to resolve the issue. And it certainly wouldn’t be the first time that the Buddha is being creative with how he phrased things.
Maybe somebody else has some more satisfying resolution. But for now, I think this text is actually most problematic for the bodily jhana view.