The bounded delay buffer management problem, which was proposed by Kesselman et~al.\ (STOC 2001 and SIAM Journal on Computing 33(3), 2004), is an online problem focusing on buffer management of a switch supporting Quality of Service (QoS). The problem definition is as follows: Packets arrive to a buffer over time and each packet is specified by the {\em release time}, {\em deadline} and {\em value}. An algorithm can transmit at most one packet from the buffer at each integer time and can gain its value as the {\em profit} if transmitting a packet by its deadline after its release time. The objective of this problem is to maximize the gained profit. We say that an instance of the problem is $s$-bounded if for any packet, an algorithm has at most $s$ chances to transmit it. For any $s \geq 2$, Hajek (CISS 2001) showed that the competitive ratio of any deterministic algorithm is at least $(1 + \sqrt{5})/2 \approx 1.619$. It is conjectured that there exists an algorithm whose competitive ratio matching this lower bound for any $s$. However, it has not been shown yet. Then, when $s = 2$, B{ö}hm et al.~(ISAAC 2016) introduced the {\em lookahead} ability to an online algorithm, that is the algorithm can gain information about future arriving packets, and showed that the algorithm achieves the competitive ratio of $(-1 + \sqrt{13})/2 \approx 1.303$. Also, they showed that the competitive ratio of any deterministic algorithm is at least $(1 + \sqrt{17})/4 \approx 1.281$. In this paper, for the 2-bounded model with lookahead, we design an algorithm with a matching competitive ratio of $(1 + \sqrt{17})/4$.