Enjoy the Sammy Nestico arrangements!
I am going to somewhat fly in the face of most of what is said here, on the basis of an assumption (well, two). Firstly, it's a conventional big band (Count Basie for example) and that there's no pa. When I was first playing, I used to play in a youth jazz orchestra and also played in a rehearsal band (it was guys getting together to play the tunes, without ever trying to gig). The experience of being among the instruments and experiencing the whole thing in such an immersive way is quite an experience.
Now, I may well shouted down a bit here but I'll press on. What is your role here? In my view, your role is, essentially, to replicate an upright bass. The parts you are playing were written for that instrument and the sound it makes. It's not about definition, it's not about sustain. You are surrounded by instruments that are occupying a whole load of the frequencies in the mid range. The job of the bass in such a band is to provide little else but bottom end (and the right notes!!).
Personally, I would think the last thing you want is some hyper rig that is all bells and whistles. An acoustic bass has none of that.
You have a Precision. Ok, Precision, flatwound strings and a simple combo should do you fine. If you sound about right when you're playing with the band then you probably are about right. The conductor/leader will tell you if you're loud. They are not going to suggest you take your wireless out into the audience any more than Walter Page would have.l Big bands tuned up and then started.
There is no need to over-complicate a simple job. Turn up, stick your combo on the floor next to the drummer, plug in, tune up and enjoy the gig.