Sample sentence: Why, then—she wondered—should he have had to carry a burden of tragedy which, in silent endurance, he had accepted so completely that he had barely known he carried it?
Let's simplify this and put it in the declarative form first:
She wondered why he should have had to carry a burden of tragedy which, in silent endurance, he had accepted so completely that he had barely known he carried it.
Question form: She wondered why he should have had to carry a burden of tragedy which, in silent endurance, he had accepted so completely that he had barely known he carried it?
The verb here is: should have had to carry, and not just have had to carry. Notice the inversion: should he have had in the question is "he should have had" in the declarative. The sample sentence is a question.
should have had is a modal usage of should, like should have seen, should have gone.
- Should he have had to do the work?
- He should have had to do the work.