If you get the same specs the Carvin will come in at $903 and the Fender will be $1399.99. Thats an Adler body, gloss translucent classic sunburst finish, 1 piece maple neck with satin finish, rosewood fingerboard with pearl dots, split P pickup with passive electronics, chrome hardware, white logo, and a red tort pickguard. The biggest difference will be 22 frets and a flatter 14" fingerboard radius on the Carvin vs 20 frets and 9.5" radius on the Fender. If you call and talk to them you may be able to get a 10" radius on the Carvin.
With the above mentioned specs I'd just get the Fender, because I like the way it looks better, and I know how its going to sound. With my own preferred custom options on the Carvin - swamp ash body, 5pc neck, maple board with diamond inlays, matching headstock, and add a MM pickup and active electronics, I'd get the Carvin.
I dont have a PB5 but I have a 2014 Carvin X54 and a 2002 American Deluxe P bass. I feel the woods and finish on the Carvin are nicer. The basses are apples and oranges otherwise so that's where the comparison ends.
For a basic build, that's Fenders bread and butter, and they've been doing it forever, so I'd go with the Fender. If you want something tailored to you, Carvin will exceed your expectations. On the other hand, if you want to save $500 and can live with an instrument thats at least the same quality, and you don't mind if it's a little different and doesn't say Fender on the headstock, the Carvin is a strong choice.