I think we have the same coupler type. This is an Atwood coupler.
Newer models of this coupler have a hole where you can insert a pin or lock, just like most regular couplers, to prevent the lever from lifting and releasing the ball.
Old models like ours don't have that.
When I bought my trailer, the previous owner showed me how he locked it. There is a hole on the side of the coupler, curb side, just behind the U-shaped part that slides forward and down to lock the ball. He inserted a lock pin in that hole, which prevents the U-shaped part (I don't know the name of that thing!) from sliding back.
I thought that was a bit of a redneck solution, but searching the web I found that this hole wasn't drilled by some previous owner, they were made like that. This hole is there from the factory, it's purpose is to lock the coupler, and it works just fine.
Here's a newer Atwood coupler, where you can insert a pin and lock it just like any coupler:
I couldn't find any pictures online for the old models, but here's mine. You can see the hole I'm talking about
You can see that when the coupler is unlocked, the hole is covered by the locking mechanism.
When there is a pin or padlock in that hole, there is no way for the coupler to unlatch: