A few years back I rode from Sierra Vista to Antelope Wells, before heading south to the Barrancas del Cobre (
Adios Estados Unidos, Hola Chihuahua Mexico ). From Sierra Vista I followed pavement for awhile, then took an old railroad bed to Douglas. From there I followed the Geronimo Trail to Cloverdale Road, which you seem to use as well.
At the Cloverdale Road intersection I turned right and followed Cloverdale Road to St. Louis Pass Road, a major road to the left and possibly unsigned, but there are signs at the intersection for Diamond A Ranch and No Trespassing / Locked Gates Ahead. On St. Louis Pass Road I rode through two locked gates ( I just lifted the bike over) and over a cattle grate with two wagon wheels. At road fork after the cattle grate / wagon wheels I turned left and followed the road to the third locked gate and the intersection with the main paved road to Antelope Wells. I made a right and it was about 4 miles to the border.
I was told by border patrol that it was fine to ride through St. Louis Pass Road, and they even unlocked and opened one of the gates for me.