It's not "just the 1st chapter number getting printed". It's the opposite: all chapter numbers are getting printed except the 1st. But on further inspection, I see that it was my mistake.
For context, I'm trying to print straight from a Paratext project, so not a module. I discovered that if I try to print whole chapters, the drop caps chapter number would appear on the beginning of every chapter as expected. But what I was trying to print is: LUK 2:1-40, LUK 3-6, LUK 11.
Naturally, the drop caps appear just fine on chapters 3-6 & 11, but not for chapter 2, because it's not the whole chapter that I was trying to print. This is where I get the impression that I wrote about in my question.
What finally working for me was to turn on the "Hide Empty Verses" option in the settings, and then just try to print LUK 2-6, LUK 11. And then the drop caps appear on every chapter.