Did Yaakov Avinu Outright Buy The Bechor?

The Vilna Gaon brings down proofs that Yaakov Avinu did in fact buy the bechor for a sum of money. We see this in the verse of Genesis (25:23), where Yakov says, וַיֹּ֣אמֶר יַֽעֲקֹ֗ב הִשָּׁ֤בְעָה לִּי֙ כַּיּ֔וֹם וַיִּשָּׁבַ֖ע ל֑וֹ וַיִּמְכֹּ֥ר אֶת־בְּכֹֽרָת֖וֹ לְיַֽעֲקֹֽב – And Jacob said, “Swear to me as of this day”; so he swore to him, and he sold his birthright to Jacob. It may be inferred that there was a new sale after the oath.