You may have to take an extra dowel and push it into each hole to press back the paint that inevitably intrudes into the hole.  In addition, the thickness of the paint on the dowel will make it tight, and may require careful sanding of the end or a pinch with plyers to get it started in the hole.

If there’s a lot of paint in the hole or if you’re customizing with a thicker turned spindle you may find it useful to run a drillbit backward in the hole to press back the grain.  Many of the turned spindles are on 4mm stock which is very close to 5/32.  As I look around my office for a non-drillbit item the right size I see the tip of my forceps (hemostat?) is about right, as is the largest screwdriver in my eyeglass service set.