I'm afraid the mustang bridge is not as faultless as people say. I have two Jags and one Jazzy, one with stock bridge and two with a mustang bridge. The stock one is way better BUT only after modifications.
I presume all that aside we intend to get your mustang bridge working right?
Firstly the rattling between the saddles: you need some sort of lubricant vaseline, lipsyl, or even grease, etc. then apply a small ammount between each saddle barrel. Tape would do it but may look unsightly.
Secondly the string jumping: there are two ways. If the saddles cannot be adjusted heightwise (old mustang bridge) then highering the whole bridge would increase string tension over the saddles. Unfortunately this would higher the action which then would mean shimming the neck... forget about it!
In this case it would be easier to file the string grooves in the two E saddles slightly deeper using a needle file. BE CAREFULL not to file too deep or you will end up with fret buzz.
My mustang bridge is a pain at times on my Jazzy, not to mention it does not truly follow the Jazzy neck profile truly.
Let us know how you get on and welcome to the boards. 😁
