Will asking for signs of the divine presence be unanswered if belief is not strong enough, but always answered when belief is sufficient? Could this be a good way, not of guaranteeing proof for the existence of the divine, but serving as a diagnostic indicator of the need for cultivating stronger faith?

This is a very deep and perceptive awareness, and speaks to the strength of your spiritual development that you are seeing the issue in this way. It is very much the case that people who doubt cannot summon a miracle. It would be going against their free will to be disbelievers, to disprove their assumptions. So this precludes the divine realm from answering a request to see a sign of the divine in their life in some way. That is not to say that we cannot entice them or amaze them on occasion to get them thinking, but that is not quite the same thing. You are asking here to have something more personal and something recognizable as coming from beyond the self and beyond coincidence. On the other hand, many, many, believers can fall short in making such requests also. It is much like the reality many prayers will not be answered fully because the one in prayer is doubting too greatly, and here it is not that they do not believe in a higher power, but doubt it is hearing them and having concern for them and is more disbelief in themselves. Both belief in the divine and in the self are needed for any divine action to occur in response to a request. So in that respect, your insight is quite correct. Rather than being proof God does not exist, an unanswered request to see some sign of divine presence is much more about the lack of preparedness of the requester, and a reason to work on this actively, to shore up what might be needed to fill the void. And then if signs appear, it will be a validation of their success in making gains, making improvements in their circumstances. This is worth striving for because so many things will come to those who are in divine alignment and who believe. Life will be easier and more rewarding. It is like any other capability that represents a potential strength, such as learning a trade, or a skill, or working to build something useful, like one’s dwelling or a career that will bring many rewards and security. Cultivating faith is no less important because this is the gateway to prosperity, and the gateway to the infinite future being trouble-free. Having strong belief in the divine and in oneself is the hallmark of emotional health and personal integration, and its lack, a diagnostic indicator that something is missing and there is work to be done to improve oneself to get back into better alignment. This will have many, many, dividends. So while this issue of seeing a sign may seem to be a small token, as a reward for being a spiritual being and focusing somewhat on that aspect of existence, it can have deep meaning for the health, inner balance, and potential future, and all it holds. So this is far from an empty self-serving exercise.