Turns out that's not quite right. Wikipedia confides that the earliest Japanese translation wasn't until 1910, a couple of decades later than I'd have guessed... Which just goes to show that plausible theories hatched in a pub aren't always spot on.
It's interesting to see the names they chose, though. These days, the book is universally known as 不思議の国のアリス, which translates roughly to "Alice in the Country of Wonders." All the 18 editions published since 1955 have had that name, and the two before that were pretty similar: 不思議な国のアリス ("Alice in the Wonderful Country," 1950) and 不思議国のアリス ("Alice in Wonderland," 1928). Admittedly, some of my translation choices here are a bit arbitrary, but they're clearly all circling around the English title, more or less.
But it wasn't always thus. Earlier, we had quite a variety:
1910 愛ちやんの夢物語 ("The Story of Love-kin's Dream") [Okay, I don't think "Love-kin" is a good translation of "Ai-chan", but it's closer than "Alice"!]
1912 アリス物語 ("The Story of Alice")
1923 アリスの不思議國めぐり ("Alice's Tour of Wonderland")
1925 まりちやんの夢の國旅行 ("Marie's Journey in the Country of Dreams")
1925 お轉婆アリスの夢 ("The Dream of Alice the Hoyden") ["Hoyden" was the best match I could find for お轉婆, but I'm open to suggestions!]
The status of the story as a dream-narrative is generally emphasised, as you can see - perhaps because dream narratives fitted an established narrative niche?
Anyway, I don't have any grand point to make, but I thought it was kind of interesting, and I'm leaving it here for later reference.