Being a mixed animal veterinarian I have banded my share of bulls. So I am familiar with what you are describing.
Anatomically bulls and dogs are not the same. If you look at a bull the testicles hang straight down. The spermatic cord/nerves/blood vessels are between the testicles and the bulls body. The sac itself narrows down around the vessels/cord/nerves between the testicles and the bull's body.
When you apply the band, the constriction pinches the nerve quickly (hopefully), stopping the pain. Constricting the blood supply will allow the testicles to die. The skin will also loose its blood supply and die. Slow death of the tissues allows the body to separate the dead tissue from live tissue without excessive bleeding.
Dogs testicles do not hang from their cords and vessels in a vertical position like the bull. They are kept in a flat (horizontal) position along the sheath of the penis. The cord/vessels/nerve are attached along the top of the testicles and cannot be separate out. The skin is as wide at the top as well as the bottom of the testicles. So almost any attempt to separate out the testicles and band them will meet with failure. The dogs testicles are much closer to their abdomens and the band can sometimes force the testicle back into their abdomen.
And do not be fooled this process is painful to any male animal. It is continued in food animals because it is inexpensive and can be done easily (if you don't count getting your head kicked at once in awhile) by anyone trained in the procedure.
I have seen dogs that have had this done. Some have gotten off easy with just an infection. Some have had to have additional surgeries to remove scar tissue, remove the missed testicle or try and reconstruct damage done by the band, to the penis and other close by structures. Some dogs die.
So please do not attempt this. Take your little guy to the veterinarian and have him neutered under anesthesia in a safe and humane manner.
If I can answer anything else please let me know.