I've updated my GDM installation to the latest version (2.4.4.3) from ports, and the keyboard no longer responds!
A bit of research shows that it is due to GDM taking over a VT that getty has already grabbed for itself, unfortunately, the directives in the GDM config file are only applicable to Linux.
As a workaround, I enabled GDM in /etc/ttys and disabled the startup script in /usr/local/etc/rc.d.
The problems that used to occur when starting GDM in this way no longer occur, so for now, this seems like the way to get it working.