Yeah, i don't love the standard Bigsby trems. They don't stay in tunes all that well. Certain ones, like the Mosrite style are similar and a little better, but overall, the Jazzmaster trem is probably my favorite, aside from ones for crazy divebomby shit, like Floyds.
The only upside to the Bigsby is that you don't need a huge hole cut in the guitar to install it.
The Jazzmaster trem needs a relatively large cutout, and the body has to be thick enough to allow the depth of the trem rout. A standard Mustang/Duosonic should be slightly too thin to allow that rout, though i know you said that yours is thicker. The standard Mustang trem is much less invasive, and works on thinner bodies. The Mustang trem is probably my second favorite to the Jazzmaster. When it is new and tight, it works well. Once it gets old and sloppy, it can be a huge pain in the dick to keep in tune and intonation. But for thinner bodies it is a good solution.
I would probably keep it fixed bridge if the only option was a Bigsby though, myself. They are okay bridges, but they don't allow much bending in the overall scheme, and especially not when you are trying to keep the thing in tune.
