Attach wall "peg" with dowel screw
Assuming that:


the screws are at least 1/4" in diameter
they penetrate at least 1" into the stud
the timber ends up snug to the drywall


... this should work fine. The key is that installation leaves the timbers with no gap to the drywall that would allow movement. If there's room for movement, the drywall will eventually be damaged and the screws could work loose. 

Yes, it would do to install the dowel screws closer to the top of the timber, say 1" down, to reduce pressure on the lower edge of the cut. You might put waxed paper behind the timbers while you thread them in to act as a lubricant and to protect the wall. 

You could simplify matters by installing each timber segment on a backer board (say a 16" segment of 1x6) using three or four 2-1/2" screws, then simply mount those to the studs over the drywall. The torque load on the timbers would be transferred to and distributed by the backer boards rather than through drywall, which is quite soft.