Ok, so take this with some salt because I have not rebuilt that particular model. I rebuilt the one shown below last weekend. It was a 6V horn originally that I am running on 12V. From what I have read, it should be fine. By inspection, I agree. For testing purposes, I would not worry about the 12V vs 6V.
This one draws about 7A when it is running. Most of the modern battery chargers have some smarts in them and they won't put out power if they are not connected to a battery. If the horn is functional, its internal resistance is probably too low for the charger.
With that out of the way, step 1 is to see if it is electrically connected inside. Measure resistance from the case at the mounting bolts to the terminal. It should be less than 2-3 Ohms. The one I just did was 0.3 Ohms.
If that measures good, use your jumper cables to connect it to a 12V battery for a quick burst (less than 1/2 second). If it makes noise, it is probably good and ready to go when you get a new battery. If not, then it is likely frozen inside. The reason for the short burst is to not burn it up if it is frozen.
The next step is to open it up and see what went wrong.
The easiest things to check are:
Does the diaphram move smoothly in the coil? Is the coil open or shorted? Are the switch contacts clean? Are the switch terminals connected to the housing/post cleanly?
If those all come back good, then it is probably just out of adjustment.
Based on the ones I have torn down before, my bet is on the diaphram being frozen in the coil or dirty/scorched switch terminals.
