If your drill press has enough depth below the table  to handle the length of the dowel, one idea that comes to mind is to drill a hole the size of the dowel  through a block clamped to the table and then change the bit to the size needed in the dowel and use the hole in the block to keep the dowel centered on the bit.  You just have to make sure that the table and block are both securely held in place.  You may need to find a way to hold the dowel so it doesn't spin

--Nathan, TX. Hire the lazy man. He may not do as much work but that's because he will find a better way.