Should I use dashes or em dashes to indicate the break and resumption of Lucy's dialog that was interrupted by her thoughts?
Yes, paired dashes would probably work better than your paired commas to surround Lucy's thoughts. (The reader might confuse those commas with the one after "traitor", and commas have a greater variety of uses than dashes. Thus, dashes make the structure a bit clearer to the reader.)
Do I capitalize "had" when Lucy's dialog resumes?
No. There is no way in which "had" could reasonably be considered the beginning of a sentence, so there is no reason to capitalize it.
How, precisely should this sentence be formatted?
When a semicolon connects two main clauses, the first clause is usually everything before it and the second is usually everything after it. Furthermore, in this case the second clause itself consists of two coordinated main clauses. I think that it's unlikely that a reader would be much confused, but why take chances? I'd probably write the sentence so:
“The unified kingdom of my childhood”—Lucy hoped to avoid discussing the reasons for its disintegration, as the children carried the blood of the traitor and she carried more than her share of the scars—“had one councilor for all the districts.”
Parentheses could also work:
“The unified kingdom of my childhood” (Lucy hoped to avoid discussing the reasons for its disintegration, as the children carried the blood of the traitor and she carried more than her share of the scars) “had one councilor for all the districts.”
Of course, because this is creative writing, you have more leeway than the author of, say, a scientific journal article. My suggestions are only suggestions.
