I was just looking at that myself. I think that for the PJ, with the Jazz Bass pickup installed where Fender put it, the standard Mustang Bass bridge won't fit. At least it doesn't look like it. There are other bridges which should fit the PJ, if that's a deal breaker. I like the stock PJ bridge. The only change I'm thinking of is modding mine so that I can make it string-through, like my American Precision was.
I'm thinking the standard Mustang bass bridge is the probably the ONLY bass bridge that would NOT fit the PJ. I could be wrong. It's been known to happen. The only high-mass string-through bass bridge I've been able to find is the American Standard mentioned above.