If you are running a club for ukulele players then it should be for ukuleles only. Banjos are not ukuleles, even if they do have the same scale length and the same tuning. The sound of a banjo is not compatible with ukuleles. Even when damped down to reduce the excessive volume, banjos have a penetrating, "clacky" quality which is quite out of character in company with ukuleles.
I can't understand why anyone would object to being asked to use a ukulele at a ukulele club. Even if their intention is mainly to play the banjo, outside of the club, it is not a big imposition to be expected to buy a cheap soprano ukulele to use on ukulele club nights. If I were organising a ukulele club, I would keep a couple of cheapies handy for loaning to any itinerant banjo players who might turn up.
John Colter.