I realize this is probably something that only I want, but just in case, here it is for Google (and for future me, who has forgotten how to do it).
Run this in the Terminal:
defaults write com.barebones.bbedit WebKitShouldPrintBackgroundsPreferenceKey -bool YES
Thanks to my pal Jim Correia for figuring that out! He asked me to name a background color in his honor, so I choose “Correia Battleship Gray”.