LOL - I don't have that problem. Any BL I played for would "rather" that I play somewhere else, therefore I'm free to choose what "I" like best.
(Posted by a lifetime member of the Lousy player with expensive gear club)
Seriously, that said, I do have a number of Jazz basses and Jazz pup equipped basses (all purchased before I got bit by the Gibson bug). However, I have yet to find a song that was originally played with a Jazz that doesn't sound just as good (sometimes even better) played with a P bass a P/J or a Thunderbird. Given the right strings, the right amp settings and right playing technique just about any electric bass can fit in the mix properly in a live setting.
To your point though, in any situation, the sound engineer can make or break you and the BL does have the final say. So as a paid player it's probably in your best interest to lean in the direction the BL prefers you lean. In which case, whip out that Jazz and flog it hard baby.![]()
I guess I'm fortunate, I don't have that problem either
I get to play what I want, how I want, with what I want. Within the limits of my (admittedly not awesome) abilities, of course. And the limits of how loud the neighbors will put up with....

(I also don't have the problem of owning a Jazz Bass
)