Attach wall “peg” with dowel screw

I'm building a cat climbing wall, and plan to have a series of large "pegs" as steps. My plan is to use 1-foot lengths of landscape timber (basically a 4" diameter log) and attach directly to the wall studs using one 3/8" x 3" dowel screw for each peg. All shelf brackets I've seen have a lower support brace and more than one screw connecting to the wall, so I'm slightly concerned that one screw won't be enough to support the cantilevered weight of a cat (~15 pounds, plus dynamic forces). Is this realistic, or will the screw pull out, or will the log put too much pressure on the drywall, or any other reason why this would be a bad idea? Should I just put the screw off-center toward the top of each log so there's more of a lower brace, or use bigger dowel screws, or will I need a proper shelf bracket to support the peg?