Adrock,
You initially had the correct reasoning. The bank requires an increase in lift assuming you want to maintain steady state flight. With no configuration changes, the only ways to increase lift are to increase AOA or increase speed. Since L/D_max for a given configuration is only a function of AOA, you can't change AOA and still keep the "best glide" condition. As such, speed has to be increased in order to make a best glide turn.
Note that the steady state condition does NOT require you to maintain altitude as a few have mentioned. It does require that you are not accelerating vertically – that is, your vertical speed is constant.
Mike