I went and measured the back of Wattgate plugs, your solid core 3/8" will not fit. If remove the flat plate and insert your solid core in WITHOUT the holding screws, your 3/8" solid core still won't fit. However, you can trim the tips of your 4awg S.C off to the shape of the "entrances", push them in and screw them down.
Last time I looked at HomeDepot, they had some commercial sockets that were pretty big. You may give themt a look.