Carroll may have been referencing that song although it is an American song, which hampers that possibility a bit. However, if the poem was composed nearer 1895, then Carroll's likelihood of hearing the phrase from the verse seems probable.
Carroll's use of the phrase seems quite different. His interpretation adds a bittersweet flavor, especially if it were composed so many years later. The persona's voice sounds wistful for the time he rowed the young ladies out ("Lingering onward dreamily") as well as for the Alice of the tale ("Still she haunts me"). Throughout, he mentions the end of things: evening, frost, Autumn, not to mention "Echoes fade and memories die").
In fact, he equates the dream with Alice herself: "Never seen by waking eyes." So a tale is a dream. When Carroll gets to his final line, he references the old song's blind devotion to believing life is unbearably wonderful, but ties to that idea that life is fleeting and story-like, perhaps due to the uncertainty of memory.