Thanks for taking the time to put in the effort to replicate and get screenshots and replay, decided to test it myself and I see what you mean. The abilities seem to come into play correctly though, as seen in this replay: http://replay.pokemonshowdown.com/gen7vgc2017-575796817 (Turn 1: Gyarados deals about 40% to Tapu Lele before Skill swap, turn 2: Gyarados deals about 20% to tapu lele after skill swap). It doesn't show Fur Coat taking effect on the stat preview, because Fur Coat isn't revealed to the opponent, so the opponent wouldn't know that Tapu Lele had Fur Coat at that moment.snip
(Note that skill swap doesn't reveal abilities when it's used on a partner in doubles, that's how it's supposed to work)
Though you should always be able to see what ability your Pokemon currently have, even after a skill swap in Sun/Moon. As you can see that by clicking your Pokemon's icon on the lower screen. So I suppose the stat preview would be more accurate to how its shown in-game if the Ability at the top would always show what ability you currently have, while it's more OK if the ability under "opponent sees:" is as little revealing as it is at the moment, depending on which abilities were revealed and not.
BTW, the link in your post leads to a 404 error, because there's a ; at the end of the link