I don't have any experience with round beads, but I would assume they would not work as well because only a small amount of the underside of the bead would be in contact with the fretboard, so the bead would be more likely to slip. As far as the length of the string between bridge and tailpiece, one-inch seems kind of short for effective use of tuning-beads. The only way to know for sure is to give it a try. Round and oval beads are inexpensive and you are not making any irreversible changes to your dulcimer by trying. Try it on one string to see what happens. If the bead holds and it can be wedged toward the tailpiece to make small changes in tuning, that's all you need.