I would actually bring my Kubicki Ex-Factor, and my Hartke VXL Attack pedal/preamp. I can dial in any sound with that.
Now, if this was a date at a prestigious studio, I'd be forced to bring my Fender Jazz Bass Plus (the one with Kubicki electronics) so that I could dial in just about any sound and when they looked at me they would see "Fender".
Its one of the only reasons I have the JBass Plus. I took my Steinberger Q4 to a session the other day because it has the piezo and they were looking for some standard sound but also needed something with a little more clack. The look on the engineers face when I took that out said it all, and this was a kid somewhat younger then me.
He fought me for about 15 minutes until I sat him down, had me put in direct and I ripped the number for him twice, once with the piezo only and the other with the EMGs only. He bounced the lines together and almost crapped at how good it was.
I told him to loose the "Fender" is the only recording bass and just listen to the player. Yes, a good studio should have something relatively standard around (I lean toward the P) in case some kid comes in with a really low end bass, however every decent bass out there can record well now.
I am always amused that the only reason Fender P and J became studio standards is that for most of the first decade of electric bass they were the only prevalent basses around so that's what most of the pro's had. I remember listening to Carol Kaye about two years ago when we talked on the phone and she told me the only basses around the studios in LA were usually Her and Ray Pohlman's P basses, Joe Osborne's J and a couple of Danelectro "Tic Tac" basses (six strings) that usually were not used for bass but were used for guitar solos and fills.
It's also usually they want you to have a set of flatwounds or a good mute on rounds (because of finger noise) even though today a good set of round wounds can really pull great sound out of many instruments.
So, I'm coming to your house with my Kubicki.
I would say if you are looking for a good bass at a decent price, go MIA Fender J Bass Plus. One in silverburst just sold on eBay for $530 and you can usually get them for around no more then $600. And they have the Kubicki electronics and Lace pickups that you can use in active or passive mode. Very versatile and gives you vintage and modern sound.