No matter what you do, a P bass isn't going to sound like an upright.
However, your playing style and technique are much more important.
You have to keep the notes short, muting or damping them almost immediately. The "chops" from the mandolin
need to be audible, and these are on the upbeats, between each of the bass notes. What you don't want is a note
that drones on. I heard an upright playing bluegrass once. The notes were very deep and long sustaining. Sounded
more like an electric than an upright. Sounded horrible for bluegrass.
The bass and mandolin are the rythym section. If you work with the mandolin player just a little bit, you should be able
to get the bass style down pretty well. It's also mainly roots and 5ths, with very occasional short runs or walks from one
chord to another.
I use an electric for working out bluegrass songs if I already happen to have one out. The playing style is very important
to getting it to sound right. The tone, not so much.
Also, I am pretty sure that you would be welcome if you have the style down, as opposed to their having no bass at all.