I would if it was all I could afford, and there's a dude like Uriah Duffy whose main bass is a modded Mexican Jazz, but probably not the way to put your best foot forward

I would get the best passive Precision and Jazz I could afford, though (yes, back in the 60's it was OK to have one or the other...nowadays you need both to survive doing sessions). I figure you either want the gig or you don't, and nobody has time to listen to your protests when the jobs have been drying up and there are 100 bassists within a 3 mile radius who would kill to be in that situation. So you either give them what they want without complaint or you sit at home jackin'. It's a very simple mathematical formula for me...Producer + cooperative musician = callbacks. Producer + righteously indignant musician = losing your phone number.
OK, so go into a session with producers and engineers who are well-ensconced in the studio world and are paying you to lay down a track and tell them that when they ask you to play a Fender. They'll have someone else within a half hour, maybe less.