This paper investigates streaming codes over three-node relay networks under burst packet erasures with a delay constraint $T$. In any sliding window of $T+1$ consecutive packets, the source-to-relay and relay-to-destination channels may introduce burst erasures of lengths at most $b_1$ and $b_2$, respectively. Singhvi et al. proposed a construction achieving the optimal code rate when $\max\{b_1,b_2\}\mid (T-b_1-b_2)$. We construct streaming codes with the optimal rate under the condition $T\geq b_1+b_2+\frac{b_1b_2}{|b_1-b_2|}$, thereby enriching the family of rate-optimal streaming codes for three-node relay networks.