Hey,
I was just reading throgh this thread and thought of a couple of things that may help.
I don’t think there is a single tone that you could pin down as ‘being J’.
He seems to have used a truckload of different effectsover the years.In no doubt an endless amount of combinations.
One common thing over the years is his use of the fatter style single coils in his Jazzmasters.Even in the Les Paul I’ve seen him play ,it had p90’s.Yeah I know he has played jaguars and Teles….
Going to a hot humbucker might send you in a direction further from the tone you are trying to achieve,driving things a bit too hard.I assume you are playing a les paul with humbuckers,maybe you could try backing off the volume on the guitar a bit or screw with the tone a little.Or you could beg,borrow something with single coils and try it through your setup.Or test drive something at the shop.
This could help,or frustrate you more.
Good luck.