William, I wish Harvey Meyer frequented this forum, he could give you the ultimate answer. But I'll give you my 2 cents from my own experience. I've tried exactly what you're asking about and yes it works, sort of. You need a small brush that has a similar shape to the brush nib tip on the Faber Castelle PITT markers. And it needs to be somewhat short and stiff bristled. Otherwise it will bend and be hard to control. But it still requires a very steady hand and I think it is more time consuming than the markers. You'll be constantly dipping. A little too much ink and it will either drip on something you don't want dripped on or it will run into a part of the wood that you don't want colored. In my opinion, the FC pens with a sharp point on the nib are better. The other 3 nibs are kind of useless on the baskets. I don't know if you're aware, but you can trim the brush nib with some fine scissors to sharpen it. And you can pull the nib out of the pen and reverse it, there's a sharp tip on the other end. In all of this, I'm referring to the Faber Castelle Pitt pens. They are india ink, most of the other varieties are alcohol based inks. The alcohol based inks and dyes bleed very easily and are difficult to control.
Also, if you burn the groove between the beads it creates a sort of shadow line between the beads. Not as distinct as a wire burns a line but more like shading it. That makes it less necessary to get the ink into the bottom of the groove (which usually results it getting ink on the adjacent bead). I burn the groove with a small piece of formica (free samples from lowes or home depot). Sand off the top layer of the formica on a belt sander leaving just the thin resinous backing, hold that in the groove with the lathe running. It doesn't take much to burn but it does seem to help with the coloring, at least for me.
Also, I only burn the outline of the areas to be colored first. Then I color those areas and burn the rest of the lines after they're colored. The burn lines tend to wear at the tip of the markers making it hard to keep them sharp so coloring first and then burning makes the marker tip last longer. Hope all that makes sense and helps.