It's generally best not to leave questions without an answer, even if they are answered in the comments. MO best practice is to post a CW answer summarizing the answer from the comments. In this case, the answer is "yes: these two constructions agree" because exactly what the OP sketched works, because $C$ is a 1-category hence automatically a fibrant simplicial category (with the discrete simplicial structure).

Aside: I stumbled upon this question just now when answering a different hammock localization question: https://mathoverflow.net/a/467080/11540. My only qualm about the present question is that I wouldn't call it a "localization in the sense of Lurie" or say "Lurie technology" or "invert $W$ in the sense of Lurie" for two reasons. First, this technology existed before Lurie. However, I agree that Lurie's book is a wonderful place to learn it. In that book, he himself points out that just because results are not attributed to others does not mean one should assume Lurie was the first to prove them. Secondly, there are several ways to invert things in Lurie's book, including passage from an $\infty$-category to its homotopy category and also Bousfield localization. So, I think it's best to be clear precisely what localization is being used, and I'm glad the OP pointed to the specific place 1.3.4.1.