The main issue would revolve around where on the flat face you needed to bore the hole(s). One hole dead centre would be different than a few holes at various locations.
In any case, Joe has nailed it. Get a piece of plate large enough to make a good solid face plate. You could turn a round piece on the back, as Joe suggests, or drill three holes in the plate to take the tips of the chuck jaws. Position the holes so that the jaws are fairly far apart. If the plate is thick enough so that the tips don't stick through, it should sit down on the step of the jaws fairly consistently. Mark one hole with a centre punch on the back so that you can place that hole on the number 1 jaw each time you mount it. Then face the plate and bore a hole in the middle of it to clear the holes you will be boring in your work piece.
You can choose whether to turn the edge of the plate round or leave it square or rectangular. It's a good idea, though, to make it symmetrical so that balance isn't a problem. On that note, bolting on some weight opposite the work piece helps to balance things out if you need to clamp the work off centre. It would be worth the time to drill and tap a pattern of 3/8" holes on the plate to make clamping your work easier.
