So many mindlessly bloodthirsty people in here.
1. He is being possessed by the ghost of an elven hero, the rescue of the elf is the only actual goal here from the very beginning. Not vengeance.
2. Others are closing in, if he gets seen by anyone then killing them would only make things worse, not better.
3. In fact, killing the prince wasn't really an option here at all since he was already seen by the cat girl adventurer earlier, whom he only knocked out and tied up. She could testify that he was on the 10th floor.
4. He needs time to clean-up before he can run, he has to collect all the arrows first (evidence removal) who knows where they all are after the shield deflections, etc.
The decision to induce maximum fear into the playboy (naive/immature) prince and avoid detection by anyone else provides the maximum upside in this scenario. The prince is unlikely to talk about the incident himself (it's highly shameful for him), and no one outside of the princes direct retinue is aware that MC encountered the prince. The only way this goes bad is if the prince himself seeks revenge, which is why scaring the shit out of him was emphasized, but again, this was likely the goal from the start. Perfectly wiping out everything in this story setting is asking for too much, readers ought not to be so greedy.