The proper incantation was to make a script in my PATH called terminal-vim like this:
#!/bin/bash mate-terminal --disable-factory -x vim -f "$@"
(gnome-terminal should work similarly if you’re in GNOME.)
and to set it as my external editor in Claws Mail by going to Configuration, Preferences, External Programs, and setting Text Editor to terminal-vim '%s'.
Then under Writing in Preferences, I ticked the box “Automatically launch the external editor” and all is well.