The Honda solenoid
might still work. All that can really does is protect the innards, but it's copped a hit somehow.
You could try a continuity check across the black and black/white wires - that will tell you if the coil is OK, at least.
If it is, then try putting 12V [e.g. from a car battery] across the same wires - +ve to black. If it's still working, you'll hear/feel it click. A full function test will need a good battery, and it does sound like the one on the mower is cactus.
The GXV390 Owner Manual has some charging system wiring diagrams, and there are three types.
Download for your model is at
http://cdn.powerequipment.honda.com/engines/pdf/manuals/00X37Z5N6111.pdf
