To start with, maybe shipment of some types of batteries should never be allowed on board any aircraft?
The problem often isn't what's allowed and what's not allowed, it's what people stuff into the boxes undeclared that can be the big problem.
I work in UPS' ground operations in Louisville (I load the planes that you guys fly) and you'd be amazed what people ship without thinking. Kids going off to college and getting their first apartment? Let's send them cleaning supplies! How about Lysol, various aerosols, alcohol? And she looooooves candles so let's send her some along with a box of matches. And while we're at it, let's send her a new battery for her laptop and wrap it all up in some nice warm blankets.
I kid you not, that was all in a box that was recently discovered because it was leaking and soaked through. Apparently the lid on the Lysol bottle wasn't tightened down and the contents spilled out. And these people didn't know any better so they didn't declare any HAZMATs.
In the thirteen years I've been at UPS I've seen all kinds of things fall or leak out of boxes. Salvaged batteries just dumped loose into a carton and taped shut. Household cleaners out the wazoo. Even snakes on occasion.
Short of opening each package I don't know how you would eliminate unauthorized HAZMATs on an aircraft. And on the night sort in Louisville we'll run a million packages or more on a nightly basis, in about four hours. Tough task for anyone.