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So apparently, a photo on Instagram from someone at Fender hints at a possible reissue of the Jag-Stang, just though everybody would like to get a look at this thing. Interesting color, looks like they are using a slug/screw arrangement on the bridge pickup (maybe an actual Seymour Duncan JB or similar this time - rather than a piss poor attempt at something else). I hope it's true and they are bringing this one out.
I tried embedding it and told me it's an "unsafe operation" whatever the heck that means.
The simple answer for this would be to either buy a new Squier or Fender from eBay or Guitar Center or wherever and slap on a new neck. My go-to guitar is an Olympic white Squier Jag with a maple fretboard, one pickup, and one volume. A lot of bang for your buck and lots of room for modding it within an inch of its life.
I think the Cobain Mustangs might have been direct mount, I got to look at some shots of one of Kurt's guitars and it looks like Kurt's own Mustangs were screwed directly into the wood IIRC (not the Compstang though).
The Jag-Stangs are not the same, the pickups mount through the pickguard. Also, the Jag-Stang shipped with a relatively weak bridge pickup originally (a 7.5K Ohm DiMarzio H3/H8 I recall) - not so sure about the "Santa Ana" pickup in comparison or if its even the same recipie done by another manufacturer.
Body mounting pickups does affect the tone a bit because it couples to the body, and on a guitar like a Japanese Mustang or a Jag-Stang that could be a great thing because the pickups couple to the body wood rather than the pickguard giving a more direct connection to the guitar's natural vibrations, and in the case of basswood guitars like these there's a lot of sonic data to pick up due to the less-as-dense body wood.
Going off of your post, sounds like you have a short somewhere involving the bridge pickup and grounding, so when you flip that switch on, it's sending the entire signal to ground like if your volume or both pickups are off, killing the whole guitar. I would look for shorts, and also make sure that the pickup is not shorted as well (ie, bridge pickup should read 11.5ish K Ohms on a Squier VM Jaguar - they are very very hot bridge position single coils - I love em').