I have a bar book from the 30's that has Between the Sheets as equal parts cognac, dry gin, cointreau and lemon juice. No rum. Quotes that this recipe is from the Bar Book from Weber at the King David.
Is that from Charles Baker? Wondrich (quoting Patrick Gavin Duffy) has the recipe as written, but I've also found sources that say it was invented by Harry McElhone (of Harry's Bar). I need to do some sleuthing to figure it out, but I'll update the drink when I do.
This came up on the homepage and I noticed your note about it being unremarkable. I couldn't agree more, and I never understood the appeal of mixing brandy and light rum. It seems like a waste of both.
I've seen a number of other drinks called a "Between the Sheets" in various old books, I think it was just a popular name a la the Corpse Reviver and the Aviation.