Rather than just lay out an answer based on what I would choose, I'll try to walk through the steps of arriving at an answer that's consistent with what you believe. I'll also give an example answer to show how this might be applied.
From the top down:
Essentially, the ethical practices that you carry out are laid on a foundation of core beliefs about what is right, or morals.
Those morals are in turn built on your understanding of why those morals are right. That is, what is the underlying ideology or set of doctrines that you hold, either expressly or subconsciously?
Now, let's start at the base and work our way back up.
First, at the ideology/doctrinal level, what constrains your moral beliefs? It could be that you base your beliefs on something changeable/subjective, like the current values held in the society where you live, or your internal sense of what's right. Depending on the culture you're in, what's accepted? Or it could be that your beliefs are based on something fixed/objective, like an eternal God who has established an intrinsic, unchangeable value of human beings.
Second, then, is what morality could be derived from an underlying ideology? If the underlying ideology is dynamic, then it's plausible to base a rule on the current social climate, either directly or on a desire to "present well," that is, whether it matters if people think you're adhering to the current social norm. It also doesn't dictate any intrinsic value to persons, but pragmatically, if others aren't happy with your actions, that could have a negative impact on yourself that you'd like to avoid. If, on the other hand, the underlying ideology in question has a static position of intrinsic worth of persons, then any resulting morality would be constrained to respect that worth, and appearance of compliance has no value.
Thirdly and finally, what ethical practice spawns from the underlying morals? For the dynamic approach, it's a matter of examining cultural norms to choose an approach, then decide whether to adhere to it, or at least, give that appearance. For the objective approach, you'd look to that fixed standard to choose practices consistent with the foundation.
Now, let's apply this to your particular case. Interestingly, walking up the dynamic side or the static side of the tree can arrive at the same conclusion, though it's often not the case. For the sake of discussion, let's say you've considered the first level, or have some long-standing beliefs in that regard, so you quickly decide to consider what's best for those impacted. In this case, that would be the recruiter and the client company, and possibly others. What's best for the client company? That might be really hard to decipher. There are many factors we might not even know of. And what's best for the recruiter? That's probably easier to understand. They make their living by connecting clients with workers, so anything that enables the process would be beneficial, and anything that undermines it would harm the effort. It's common for recruiters to withhold the client name in order to protect their ability to complete the sale, as it were. They might give out a name as a result of naiveté, or because they trust you.
So, you might decide not to bypass the recruiter for a couple reasons. One, the recruiter who's trying to help you, and in exchange gets paid when they do that successfully, is now negatively impacted, and you choose not to do that. Or, you see that you've been trusted with information, and may want to work with that recruiter again, so choose not to sever that future relationship, because it's not in your long-term interest. Or, you decide that the recruiter has more leverage to negotiate your salary or provides other practical advantages, and you decide you want to benefit from that.
All in all, it could be a straightforward question, but it's complicated by various issues, both ethical and pragmatic.