So I got *lucky* and bought a Classic J5 from a TBer, and we had a nice meetup. It's definitely a very nice bass. The tone is very even all over the neck, and the B is nicely integrated with the rest of the strings. It's bright, but has a vintagey tone. The G string and higher notes in general are nice and thick, something that isn't always true on jazz basses. There's a slight dead spot around the 4th fret on the G string, but I wouldn't hesitate to use that note in a bass line, it still has punch.
I think I'm going to change the strings, it currently has brand new Stainless Dunlops on it, and they seem a bit clean and hifi for my taste. My current favorite strings are the Cobalt rounds. I'd like the bass to sound more aggressive, a bit more P-like in the mids, and I'm hoping the those will do the trick. The neck pickup is already more P-ish than other jazzes I've played, so this bass could give my Nordy P5 a run for the classic/hard rock stuff if it likes the Cobalts.
Here's my current family, I think it fits right in:
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Congrats you have some killer basses. Your tonal findings are in line with mine. The Moollon has such a rich and thick J tone - really like the good ones should and do sound.
I'd strongly suggest trying the Fodera nickels. They accomplish what you are going for tonally. Thick and warm in the mids. When I really want to get aggressive. I turn on my 5TH gear OD pedal barely past 9 o clock and run it through a Tube pre. It's easily the best bass tone I have ever had.