Yikes! It sounds like you have more than a pack rat chewing through a wire, causing an OPEN. You probably have a +12 VDC short to ground. Just guessing, but probably somewhere, a +12 volt line is laying across a metal pass-through hole, and the insulation has rubbed through or been cut so that the center conductor is shorted to the chassis (ground).
First, disconnect your battery completely, both positive and negative cables. Also, disconnect from shore power so that your converter is not providing power to the +12 volt and DC ground lines.
If you have solar or any other 12 volt
DC sources (e.g., a battery charger), make sure those are also disconnected. Remove fuses, switch circuit breakers OFF, etc., so that the positive and negative 12 volt
lines are completely dead.
Next, I would start with the easiest things first. Visually inspect the +12 volt battery cable, beginning at the battery terminal connector (which should NOT be connected to the battery during testing), and trace the cable back through the battery box and the through-hole where the positive cable enters your Airstream. Look at the insulation at cable clamps, pass-through holes, etc., where the insulation may be damaged and bare conductor wire is visible through the cut. This cable should go to a buss panel where wires start fanning out to the rest of your Airstream. In addition, look for sheet metal screws and other metal objects that look out of place, which may be stabbing through the insulation and connecting the inner conductor wire to chassis ground.
Note: Think back, since the last time your batteries were OK, have you drilled or cut any holes or screwed sheet metal or wood screws into the walls, floor, etc.? Have you done anything that could have cut or stabbed through hidden wires? Make sure to check these locations first.
Make sure all of the connections on the buss panel are clean and tight. Insure that stray wire strands, extra lengths of bare conductor, mounting hardware, metal twist ties or parts, set screws, etc., don't protrude and touch other bare wires and connections. Also insure connections are secure and none are free to move around, causing an electrical short to adjacent connectors.
Follow the biggest, thickest +12 volt wire (probably has red insulation) from the BUSS panel to the FUSE panel, which is most likely located near your converter. Check the cable for cuts and scrapes, etc., as above. Then, check the routing and insulation on any other +12 volt wires that come off of the buss bar, and follow them all the way to where they terminate at a light fixture or appliance. (Probably, there are only a couple of +12 volt wires; and the biggest one will go to the fuse panel.)
If you haven't found anything yet, remove all of the 12 volt fuses from the fuse panel (make sure to keep track of where they go for later reinstallation). One end of each fuse goes to a light fixture (or several lights/appliances connected in parallel on that line), and ultimately ends up connected to chassis ground. The other end of the fuse goes back to the +12 volt connector on the buss panel.
Using an Ohmmeter set to low resistance (e.g., Rx1, Rx10, etc.) with one test lead connected to CHASSIS GROUND, probe the electrical connections at both ends of each fuse holder.
Note: A continuity "buzzer" would also work; and it might even be better, if you are working alone.
One end of each fuse holder goes to the +12 volt BUSS PANEL; and this end should NOT indicate a low resistance SHORT to chassis ground. If any of these do, THAT WIRE OR SOMETHING ON THAT CIRCUIT IS DAMAGED, AND THIS LINE SHOULD BE THOROUGHLY INSPECTED.
The end that goes to the light fixture or appliance may indicate a direct short to ground, which is expected. However, it may also read as an OPEN circuit; because the ON/OFF switch may be in the OFF position.
If no short to chassis ground is found on the +12 volt side of each fuse, I would begin checking the wires that go FROM the FUSE PANEL to light fixtures, appliances, etc. Use your Ohmmeter to check the second side of each fuse for a SHORT to CHASSIS GROUND, and start with the circuit that has the highest rated fuse, since you indicated that the negative cable "exploded and burned". (The circuits with smaller fuses would probably have destroyed the fuse before burning up your cable.)
All of the above assumes that test readings are not affected by Ohmmeter test voltage leakage through your converter. Without actually trying this, I'm unsure if this will be a significant factor. However, even if all readings indicate some resistance, you may be able to make a good guess at which circuit is shorted by the Ohmmeter reading closest to ZERO Ohms (the lowest resistance). Mostly likely, any leakage back through the converter will have significantly higher resistance. If not, you may need to disconnect the +12 volt cable that runs between the fuse panel and your converter's +12 VDC output.
Once you isolate the circuit that has the short, you just need to carefully inspect those wires for damage. Also, look for burned wires, bad splices, overheated switches, etc. in that circuit.
The above is a best guess at how to approach troubleshooting this problem, as it's hard to do this without actually being there.
Please post your inspection and test results, especially if you don't find the problem source. Your results may help indicate what to look at next.