dip some water out of a bucket and the water comes from the bucket.
Run a hose into the bucket and dip some water out of the bucket and now where does the water come from, the bucket or the hose? Hard to answer, same with the battery/charger combo. It comes from where it is connected to. It is all connected together.
In your case, no the AC is never part of the DC circuit other than the charger runs off AC power.
Sounds like you blow a fuse on just battery as well as with the charger/converter on.
I think the question got unnecessarily confusing.
LEDs must be bad (not sure how), or a short in the wiring. More likely the short is at a connection more than where a wire might have gotten damaged. Check connections first.
Check that the LED lights work. Connect them directly to a 12 volt power source and see if they work. Connect a fuse inline and see if it still works.
If all is good then you probably do have a short.
Finding electrical issues like shorts is not fun and not easy. There is no good answer at this point.