If it can run on one cylinder, that means at least one out of the two circuits on the igniter is functioning. That means the bike should run on at least two cylinders. 1 & 4 are a pair, and 2 & 3 are a pair. If one cylinder of a pair is sparking, the other should as well. If the coil has a problem or a plug or plug wire has a problem, one cylinder of a pair might not fire, but this symptom cannot be caused by the igniter. If that circuit works for one cylinder of a pair, it is also working for the other cylinder of the pair.
So maybe check plug wires or swap coils.
Hooking up the reg/rec backward wouldn't necessarily affect the igniter. That should only affect the reg/rec, battery, and associated wiring, which is usually mostly limited to the ground wires and the reg/rec output wire to the battery.
Regarding that, there should be no ground wire (black/yellow) connected directly to the stator, but you mentioned one. Perhaps that ground wire is bundled with the stator wires, but it probably goes to something else. However, if it melted, there is a chance it melted some stator wiring along with it, so check all of that carefully.
None of that should really have affected the ignition unless there were some ignition wires bundled with reg/rec wires somewhere and it melted. You may have some serious melted wiring issues to look for.
I would pull off all of the ignition parts and check them off the bike, that way you can eliminate the wiring loom as a culprit. This diagram is for a 550, but the 1000 would basically be the same. Tapping the pole pieces on the pickups with a steel screwdriver should cause the coils to spark.
Note: make sure each coil has a path for the spark to go from one plug wire to the other. Failing to do that can damage the coil and/or igniter.