Congressman Bunner died ca 1837, so forget him at Lincoln's deathbed; however, if there is a Bunner or Heath (another sister) present I'd like to know. You said there was no Tuckerman in your Reck book, so that eliminates Henry Tuckerman, brother of Charles K. Tuckerman, and I'm just trying to think of someone C.K.T. would trust to get the real piece of sheet. That was an unintended possible play on words, but it seems appropriate.
Nope, there was no one present at the deathbed named Bunner or Heath. I am of the opinion that if the sheet were obtained by someone, it was not a person at the deathbed. Edwin Stanton would never have allowed the taking of a such a "souvenir" that sad morning of April 15, 1865. As far as I can tell the Petersen family was left with the bed sheet; what happened from there I do not know. As I mentioned before, Petersen descendant Robert Bain writes that William Petersen said, "We will give Mr. Clark some new bedding." However, Bain's book is a "family lore" book, not a history book with footnotes, endnotes, bibliography, etc.