I hope you have told the broker "NO" in the most emphatic terms, sprinkled with a few spicy adjectives.
The agreement of sale was signed by both parties, and the agreement was executed by the title company when they transferred the deed to you. To come back later and claim they can't close until a month later because they have an existing reservation is a change to the contract after it has been executed.
The deed is recorded, so the contract is yours. The prior owner doesn't get to use the contract any longer, regardless of what points are in it, or where they came from.
If you decide to give them the extension, ask them for $1000 (or more!) for the late closing date as consideration.