As far as I can tell, there was no serious objection to Dalton's atomic theory in the sense that there was no competing theory going around. Rather, some scientists cautioned that the evidence for atoms was all indirect, and the believer might have been taken in by an illusion.
In the 19th century there were two forms of evidence for the existence of atoms and molecules. The first was the kinetic theory of gases, which mathematically modelled the behavior of gases (pressure, temperature, diffusion, etc.) on the assumption that gases are made of particles. The second was Dalton's law of multiple proportions: chemical substances combined and broke down by weight in ratios of small integers. For a good century this was all there was. Then in the early 20th century, Albert Einstein and Jean Perrin investigated Brownian motion. Einstein developed a mathematical model, then Perrin proved that Einstein was right in some experiments. This third form of evidence was enough to convince the doubters that yeah atoms and molecules are the truth of reality.
Is my narrative correct? Am I missing anything?
I think when you start with Dalton you're already sort of tilting the narrative a bit in one direction (towards acceptance). My sense is that many of the "real" battles that set the stage for Dalton's atomism were those of the 17th century, and the techniques/refinements that got built onto them in the 18th. These included the discussions about the nature of chemical substances by people like Boyle, and the victories over Aristotlean plenism (anti-vacuumism, which is also anti-atomism). Once you have something like the "chymistry" of Boyle being reasonably accepted (in which there are many different types of matter and elements), you are in a place where something like the Daltonian "chemical atom" starts to make a lot more sense as a way to describe compound substances. You also, in the same time, have things like Newtonian atomism already being put into some kind of use.
As you have it, the discussions after Dalton were about how physically real one might want to speculate atoms were — whether they were a heuristic, or just a mask of some other misunderstanding, however useful they were for chemistry. My sense (I keep saying "my sense" because there is no doubt a lot of complexity here and I am generalizing from bits and pieces as opposed to some authoritative account) is that as you put it the 19th century work on thermodynamics simply does not make a lot of sense without assuming the existence of physical atoms, even though their nature was very much unclear though speculated on. Vortice theories, for example, date from this period, as does Prout's hypothesis and, eventually, Thomson's all-electron model of the atom. All of which would lead, in the early 20th century, to the Rutherford atom, Bohr atom, and so on.
Which is to say, I think putting emphasis on Einstein and Perrin is making a common error in the history of science, which is to look at an explanation that was given to justify what almost everybody already believed as the reason that people believed it. By the time Einstein is writing on Brownian motion, if you are not a believer in atoms (for whatever reason — the main one I know of is Machian suspicion at a lack of direct observation, and that is more a methodological argument than an ontological one, in that Mach was saying you couldn't admit the idea into the realm of science, not that he positively did not believe in atoms), you probably won't be convinced by Einstein's argument (which is still an indirect observation by those definitions, for example). But such people would be old and rare anyway.
I would be very surprised if there was anyone in the early 20th century who was a doubter of atomism who was convinced by any evidence at that point. Scientific "convincing" of non-believers is very rare and not the normal mode of how the scientific community functions; usually it is more a form of "displacement" than "conversion." As Max Planck famously decreed (and Thomas Kuhn made an entire theory of science basically based around the idea):
A new scientific truth does not generally triumph by persuading its opponents and getting them to admit their errors, but rather by its opponents gradually dying out and giving way to a new generation that is raised on it. … An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning: another instance of the fact that the future lies with the youth.
One sees this in other areas of the history of science as well. Heliocentrism is probably the most obvious example: the gradual growth of heliocentrism among astronomers was not due to some new discovery or better data or anything like that, but other (arguably "non-scientific") reasons (like the philosophical, aesthetic, theological, etc.). By the time (in the 19th century) that experiments could actually definitively confirm heliocentrism (the measurement of the stellar parallax, for example), it was already something taken for granted by anybody who mattered in the scientific community, and if you were one of those rare fringe people who still didn't believe in heliocentrism by then, you probably weren't going to be convinced by new evidence.
But there is much missing in my little version of things, too (it bounces far too readily from one big name to another, and in my experience the history of science is much more fine-grained than that, with lots of little disputes and debates that are lost to all but the very careful historian), but this is my sensibility on it, from what I've read on atomism, thermodynamics, Mach, etc.