You are right in regards to the conflict and resolution. What you need to ask yourself in determining the climax of any work is, "At what point is the conflict at its highest point--in a make-or-break situation?"
In the case of Anna and King of Siam, Anna must accept after the execution of the lovers that while the king admires freedom and President Lincoln, there are some aspects of his culture that he believes that he cannot change immediately. The execution serves as the climax, because it forces Anna to think about not only her own inner conflict but also the inner and external conflicts of the king. She must realize that she can effect change in Siam but that it might not be in the timing that she desires.
A fictionalized biographical history such as Margaret Landon produced with publication of Anna and the King of Siam is certain to feature numerous conflicts between cultures. Indeed, the book's opening passages depict the juxtaposition between Anna's fellow travelers on a Siamese steamer and Anna herself, a privileged but sensitive woman who arrives to tutor the king's son (and heir to the throne) and other indigenous children of privilege. Describing the "troupe of circus performers" and snarling dogs that populated the steamer, Landon soon introduces the reader to her protagonist:
"Somewhat apart from the rough and laughing group an Englishwoman was leaning against the rail. Her dress of lavender mull had a neat high collar and modest wrist-length sleeves. She was slender and graceful as she stood there with a light breeze ruffling her full skirts. Chestnut curls framed a face that was pretty except for the rather prominent nose. Her dark eyes were turned toward the line on the horizon that was land. She stood almost motionless, fingering a curious brooch on her breast, a gold brooch into which were set two tiger claws."
As Anna and the King of Siam progresses, this juxtaposition—the enlightened Westerner introduced to the less-developed native population and its powerful but culturally backwards monarch—inevitably illuminates the vast gulf existing between the two cultures. Landon's story does not lack for examples of conflict that lead to a climactic scene. A product of the British Empire, Anna is suddenly confronted with endless examples of the barbarity and castle intrigues characteristic of the Siamese Royal Court. Regaled by stories of the poisoning of the "Second King" and the ethnic and cultural clashes that dominate the reigning monarch's harem--Lao women versus Siamese women, for example--Anna is able to attain the full measure of King Mongkut's rule. This is, after all, a more primitive society, and its notions of justice are considerably different from those of the English world of jurisprudence from which Anna has emigrated.
What makes Landon's story particularly interesting--and, it should be noted, the story's veracity has been widely questioned by historians and biographers who note Leonowens's tendency towards embellishment and outright dishonesty with respect to her personal history--is that Anna has been invited to the kingdom for the express purpose of facilitating the opening of this remote kingdom to Western influences. Anna's principal function is to teach the king and his multitude of children the English language while enlightening the monarch on Western ideals. This makes the cultural clashes between those Western ideals and the king's normal practices especially dramatic. Anna is tasked, essentially, with helping the king to modernize his kingdom, but she is forced to observe the barbarity that is integral to King Mongkut's rule. A climactic passage, then, could well be the description provided Anna of the banishment of the concubine Klip and the fortune-teller Khun Het Na, whose machinations were blamed for the assassination of the Second King despite speculation within the Royal Court that they were merely scapegoats. As Landon describes the scene:
"The concubine Lady Klip, the fortune-teller Khun Het Na, and nine female slaves were tortured and publicly paraded through Bangkok.... Afterwards, they were thrown into an open boat, towed out to the Gulf of Siam, and abandoned there."
This charade perpetrated by the king casts absolute doubt on his character, as it is made clear to Anna that the death of the Second King benefited King Mongkut and the fact of poisoning had been kept secret because of the shadows this information's revelation would cast upon the monarch. The executions of the two lovers may constitute the novel's climax. That act betrays the king's claims to legitimacy as much as the suspicions of his role in the Second King's death. Whereas the young lovers' deaths represent the lengths to which the king and the kingdom have to go before the king's vision of a modern Siam is realized, it is the machinations surrounding the poisoning that undermines the king's legitimacy and can be considered the climactic passage in Landon's book.