With a rotary attachment a laser might be able to handle that, but there are severe design limitations. With a 3/16 dowel you have less than .6 inches circumference; 42 point type would wrap all the way around. I lasered some standard flat sided pencils, they should be about that diameter, but I did them flat and the type was in the 5 point range.
I haven't looked it up in the manuel for the rotary attachment, but the it would need to turn a long way around to do even small type. I use the rotary attachment to engrave on the outside of champaign glasses and crystal coffee mugs. In those applications the rotary is turning about 1/4 to 1/3 of a turn on each side, my rotary will not turn enough to spin the glass around and engrave the other side, the glass must be removed and repositioned in the clamp.
For a 3/16 dowel you would need to make more than 1/2 of a full rotation just to apply two lines of 9 point type.
The laser will easily burn through the paint, but it might leave it looking scorched around the edges. Put a piece of masking tape over the paint, burn through the tape, the paint, and into the wood and the scorched edge will be confined to the tape that you can peel off and throw away.
