I'm not sure about the Dolphin Nose model, but I used an older styled JJ as my main bass for years. I had (and still have) a single cutaway, short scale bass with the coke bottle headstock (matching color, saWEEEEEET!) and Neptune bridge. It was my first "real" bass, and even custom ordered it years ago. It was a lot of fun to play, and had a pretty fast neck. It was a bit neck heavy with the wrong strap, and I always felt the neck was much nicer/solid than the body, and occasionally thought of having a wooden body made for it. Oh, the neck has a very flat radius.
I recently listened to a live recording I made with that bass, and never really realized the distinctive sound it has. It's very punchy, and really cuts through the mix. I mostly used to use both pickups together, as they tend to be weaker soloed.
In the last few years I moved back to full scale basses, and haven't really looked back. Coincidentally, I just installed piccolo strings on it, but haven't played it too much yet. The tuning machines have small string holes, and it really restricts your choice of strings. You need a 30" string with a fast taper, mostly for the E. There's a list of compatible strings on the website.
D'oh! I just remembered that I also used to own a JJ single cutaway guitar years ago as well as a similar model Dano re-issue. I found the JJ to be far superior to the Dano, and I quickly sold it off when I bought the JJ.
Lastly, I've emailed back and forth with Jerry a few times over the years. He's a nice dude, and has been friendly and helpful.
Hope that helps.