i propose that sleep tight simply means sleep well or deeply; thus, deriving good-rest. The word tightly, although not often used in this way now, means 'soundly, properly, well'. The earlier phrase 'tight asleep' derives from this meaning, as seen in this example from Marie Beauchamp's novel Elizabeth and her German Garden, 1898: "And once, when there was a storm in the night, she complained loudly, and wanted to know why lieber Gott didn't do the scolding in the daytime, as she had been so tight asleep."
Good answer, but the truth is that this term derives from the fact that many early Americans slept on beds of rope covered with straw. If the ropes loosened, good sleep was impossible. thus the term "Sleep tight".
Where I learned about this was during my tenure as a guide in Custer State Park at an old fort. I researched it no further but think locale has much to do with the meaning. Your research is correct but so is history!