Standard 2x4s will hold wood screws just fine. Use a screw with the largest gauge that will still fit through the bracket, and make sure it's at least 1.5" long so you have more threads in the wood.
But instead of mounting the 2x4 on the outside of the wall, since you have access to the stud area behind the wall,why don't you put lengths of 2x4s between the studs behind the wall -- either toenail them into place or double up the studs and secure the horizontal pieces to the new vertical pieces -- and then screw into the new horizontal 2x4s from the front?
