Jere,
The book file doesn't care if you call the component files chapters or they're just portions of a chapter. A chapter in the book can include as many files as you want. Also, you're not using the book file to set up numbering, which is causing your problem.
The easy way around is to make each of your "sections" a discrete file with the appropriate size master pages. Add them, in order, to the book. Select all of them but the first, right click, select Numbering > Chapter > Use Same Number As Previous File. Then click the Paragraph tab and but select Continue Numbering From Previous Paragraph.
Then select the first file in the chapter, and do the same thing, but set it to Continue Numbering From Previous ... so that it increments from the preceding chapter. That sets your chapter and para numbering for all the component files.
Save, then Update.
Art