You've hit on something I consider a real mystery.I don't know who Esther Beard's parents were.However, it is interesting that Lewis Floyd was a surety for Jacob Beard's pension application, and that one of Lewis Floyd's sons (George) married a daughter of Jacob Beard and his second wife, Rosanna Wiseman/Windle.
I have only two pieces of information that might help with Esther Beard.(1) Jacob Beard first married Mary Stockslager.Mary's mother, Esther Reese, left property in her 1818 will to her daughter Mary Beard who then died about 1819.This property was eventually distributed to the chidlren of Jacob and Mary Beard, but no Esther Floyd is mentioned in the various settlements - only Isaac, Joseph, Jacob and Elizabeth.(2) I also have a letter written in 1854 to one of Jacob Beard's grandchildren that says Jacob Beard had three sons and two daughters by Mary Stockslager.The two daughter's names are given in this letter as Elizabeth and Mary.Elizabeth, wife of John Doll, was a recipient of the Esther Reese legacy, so the letter appears accurate in that regard.Mary was not mentioned in the settlement (as far as I found) but a Mary Beard was among the puchasers at an estate sale of Esther Reece.
Botom line is I can't prove one way or the other that Esther Beard Floyd was a child of Jacob.She doesn't appear to be a daughter of any of Jacob's brothers either.
Contact me outside the forum if you want to discuss further, or if you want what I have on this family.