Turns out that bbedit uses the standard system terminal. Someday I will figure out how to subvert that because I cannot imagine any reason that I would ever want to use Apple's dumb old terminal program when iTerm2 exists. In the meantime, I asked the lovely people at Bare Bones Software how to change bbedit's behavior.
Turns out it's right there in the Expert Preferences list (about a hundred obscure things that I never thought of as I searched the Preference preferences for some way to control this). To make it easier for future generations, I offer the complete command line:
defaults write com.barebones.bbedit TerminalBundleID -string "com.googlecode.iterm2"
It works and make bbedit another fraction better.