The company "Chariot" offers child strollers suitable for athletic use, including models that convert to a bicycle trailer.
The bicycle trailer hitch uses a ball and socket. The socket appears to be a stainless steel casting. It captures the ball from falling out the rear, and uses a vertical pin to keep it from pushing out the front.
The pin is removable to detatch the trailer: unhook the rubber strap from the bottom of the pin and lift the pin out of the way, then the ball slides out the front. The pin is held by both gravity and a rubber strap that resists lifting.
With Breeze-out style dropouts, the pin hits the dropout before it comes free, and cannot be removed. You can work around this by releasing the quick-release and removing the rear wheel, then removing the pin. That markedly reduces convenience.
It appears there is enough clearance that instead of lifting vertically, the upper hole could be slightly outboard and the lower hole could be slightly inboard. It appears this would both maintain structural integrity and provide clearance for Breeze-outs.
Cyclists with Breeze-outs probably want to write before purchasing a Chariot, and find out if this problem has been solved.