I've been playing bass for over 35 years now (dear god, is it that long?? ) and have done [i]many[/i] auditions, for big names and unknowns alike. The criteria that a band or artist uses to judge whether an auditionee fits the bill are many and various, and can be bigoted, pointless, illogical, and sometimes downright rude. I've discovered that over the years I've been rejected as soon as I set foot in the studio and without even playing a note because of my colour, my clothes, my bass, my hair (or lack thereof) my musical tastes, but never my playing as such. They may not have liked how I played it, but it was never the case that I couldn't have played what was required.
The dynamic between a band and a potential new member is a difficult thing to quantify, but ultimately they will choose what they [i]think[/i] they want, whether it's actually what they [i]need [/i]or not, and there's not a lot you can do about it. Also, if you misrepresent yourself, it will all come out in the end. My own impression from what you've said is that they could be "unhelpful and over-controlling", but equally by resisting a request to learn new songs, they will think you are the same!
You don't know them, so give them the benefit of the doubt. They might be great people, an are probably not relishing the task of auditioning any more than you are. Learn the songs and show willing, and if you don't hit it off when you meet, no great loss. At least you'll know some new songs!