I read this again a couple of times Peter and I think I see what you are trying. The floor standing trophy finally clicked. You will be using at least a 3/4" thick base so if you drill the holes with a good brad point bit or a Forstner on your drill press the holes should be tight enough to keep the dowels straight while the glue dries. Since the bottom won't show I would drill all the way through. You didn't say how thick the upper board will be but I would drill it as deep as possible.
Your biggest issue will be lining all the holes up between the two pieces so I would drill the holes in the correct positions through a scrap piece first and then use it as a guide to drill the other holes by clamping it to the pieces you will be using for the stand. That way all the holes will be in the same positions. I would glue it all at one time and then check to see if it is square. If it's not perfect a little twist or push should straighten it out and it should dry straight.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.