Don't thread the breakaway cable through the chain. Think about what it's for. If there is a catastrophic failure of your hitch and chains, the break-away cable should stay attached to your Tow Vehicle as the trailer starts making it's merry way down the road without you. That pulls the pin out, and engages the brakes on the TT, reducing the distance it will travel AND the number/likelyhood of hitting other cars as it careens out of control. It may also limit the damage to the TT if it breaks loose.
So, it should not be threaded through you chain, but separate from them, in a straight line from the TT to the TV.
It's OK to hook it into the same place the chain hooks on that side, just don't hook it to the chain itself.
We use one of those screw-locking chain links to easily attach it to the place on the hitch where the chain attaches. On our last TT, we would just pull the pin, loop it through the chain receiver and then reconnect the pin.
And, I think lots of people pull it accidentally once in a while. I was coming across a big dip in the road to park my TT last time and it got caught and pulled out. Couldn't figure out for a sec why the trailer was so hard to pull all of a sudden. Thankfully it hit me what had happened before I pulled it more than a foot or two with the brakes locked down, and it was across grass. And, I know now that that emergency pin really does work