Grandma's Oak Bucket
The Song:

In order for it to feel more authentic, Beck filled out his Song Reader book with fragments of songs and melodies, as well as fictional album collections and song titles. "Grandma's Oak Bucket" can be found on the album, Hymns For Him.

While a lot of the song titles on Hymns are made-up, some are references to older sheet music/songs. I was unable to find a song with this direct name, though there are a number of sheet music songs about "my old oaken bucket" or the like, so I believe "Grandma's Oak Bucket" probably did happen.