First, there wasn't a lot of strategic logic in the timing of the bombings. The rapidity had more to do with the weather conditions than it did any big plan to convince Japan of anything. Originally they were meant to be spaced out a week apart. It also was not at all clear to anyone that "two" would be the magic number (assuming it was); Groves and others thought maybe eight or so would need to be dropped. So the retroactive "we did this because we wanted to demonstrate X" story is very much an after-the-fact justification for dropping two bombs with almost no deliberation time between them (and it is even more compressed than it looks if you take into account when the Japanese even first heard the US claims that it was an atomic bomb at Hiroshima, much less when they were able to confirm the claims weren't fake — they basically learned all of this the night before the Nagasaki attack).

As for what the Japanese knew about the US capabilities — basically nothing. One of the tightest secrets of the whole project was how long it took for the US to make more bombs. But a large part of the post-Hiroshima newspaper stories (and propaganda) was about exactly how large the US production system was, which in turn would lead one who was paying attention to reason that the US did not have just a one-off weapon. It should also be remembered that the Japanese were not ignorant of the theory of nuclear weapons; they just thought that no nation would be willing to commit the resources required to pull it off so quickly.

But I just want to push back on another myth that I think is latent in your question: the idea that the US was "bluffing" about being able to drop more nuclear weapons on Japan is false. The US had a production system that could regularly produce atomic bombs. They had a third one that would have been ready to use about a week after the Nagasaki attack (August 17 or 18). Their production system could, if it was going at full-effort, produce about 3 Fat Man style nuclear weapons per month, and 1 Little Boy weapon every two months. If they had done the work to develop composite (uranium+plutonium bombs), they may have been able to double that production.

So if the war had gone on, and the US had wanted to, they could have destroyed Japanese cities on a fairly regular basis. Or, as they contemplated, "save up" the bombs to use for other strategies (e.g., as a way to aid in an invasion, by clearing out a beachhead before sending US troops through). The idea that the US "only" had two bombs is a myth. It had two bombs ready to use in the first week of August 1945, but it had a bomb-production system that could produce many more bombs as time went on.

Lastly, it is worth noting that scholars are not very sure that the atomic bombings are what pushed Japan towards surrender. There were several factors in play at the same time, including the Soviet declaration of war and subsequent invasion of Manchuria on August 8/9. The Japanese Emperor did find it convenient to cite the atomic bomb in giving his reasons for surrender, but that might have been about saving face (that there would be more honor in surrendering to "the power of the universe" than an individual man, as one American participant suggested). It is not clear that imaginations about the US stockpile played a big role in general.