| Many of us with JC's have done either or both of the following: (1) Swap the tuning machines out for a set of Hipshot Ultralites, and/or (2) add a "mod bar" to the bridge. The latter is designed and sold by a guy named Evan Webb, and if you snoop around here on TB you'll find plenty of threads that discuss this little gizmo. Add a nice non-slippery strap, as you've already done, and your neck-dive problems are gone. In addition, the mod bar has a couple of other benefits besides adding weight to the body: The extra mass added to the bridge adds a little more sustain, and it increases the length of string between the ball and the bridge saddle, which is important with some brands of strings to ensure that the good part of the string is on the saddle.
I did both of these mods and am very pleased with the results; they've made a great bass even better. I also saved a few bucks by buying the tuners used here at TB, so keep an eye on the classifieds. I think I saw a set there recently, actually. Good luck, and enjoy! | 
