Back wheel jumping out of rear dropout with child transporter
The green anti-rotation washer is at least part of the problem. Its purpose is to prevent the nut rotating easily , but with the bracket in between, it can't, and any torque on the bracket is transferred to the nut. 

I expect you'll have better results if you put the washer outside the bracket; this works on mine with QR skewers. Even omitting the washer might be better than what you've got now. With an extra part in there you may need to do things up a little tighter than normal in addition.