The only way to know who Tyre Bolsom Mauldin Hawkins was is DNA. If he was illegitimate, it will give you some leads. If he was a Hawkins, he will match other members of the Hawksins family. You definitely need the yDNA.
Edith, I have asked for more information about the deed. The copy does not say Benjamin Hawkins, Jr. Just because it was sold after Benjamin, Jr's death does not mean it was his property. It could be property of Benjamin, SR, that was not settled before Junior's death. If so, Benjamin Jr's wife would sign as heir. Just another scenario. Do you have something that shows which Benjamin the land belonged to?
I believe the name Tyre Balaam Mauldin Hawkins came from Sarah Mauldin. I believe her to be the sister of Tyre H Mauldin and I believe that Balaam Mauldin was also a brother of these two. So Tyre Balaam would be Sarah Mauldin's two brothers's names combined, a combination also used in Balaam Mauldin's family. Tyre and Balaam are both believed to have died around 1810-1811 while migrating to KY and TN. Their wives both returned to SC by 1820.
I am not related to your Hawkins, so I don't have any family story passed down. If his grandmother told him his father was not a Hawkins or that she was born a Hawkins, then believe Grandma. But if she refused to talk about her family, it does not necessarily mean her son was illegitimate. Do the DNA it will answer that problem.