Hmm. I think the US had at least one more core for a fat man type weapon.
Specifically, the demon core[1]. It was the core-in-waiting for a third bomb if needed, and I believe that the uranium and plutonium production lines were ramping up by that time[2].
[1] https://en.wikipedia.org/wiki/Demon_core
[2] https://www.atomicheritage.org/history/timeline search for 1945 July 23 entry.
40 years ago I wrote a paper on nuclear weapons. My memory isn't perfect and the sources are 40-50 years out of date. But here goes.
The US built a massive manufacturing pipeline to make nuclear weapons. Little boy was a once off. Fat man was basically a functional prototype. And the manufacturing pipeline was running and kept running after the war. How much weapons grade plutonium they had left over after they bombed Nagasaki is kind of a quibble.
There only existed two bombs at the time. Had the bombs been dropped as a show of force, it is very likely the US and Japan would have to suffer through a costly invasion anyway.