I can only speak for Switzerland, but I guess this is regulated equally in other countries.
The radio license you need will be issued by the association of your country that represents musicians (those are usually private, but have the right to collect money).
In Switzerland this is SUISA, in Germany it's GEMA and in the US it's the ASCAP.
The thing is, that when you're streaming music, you're making a public performance and this entitles the musician to receive royalty fees for this performance. Since it would be close to impossible for every band or musician to collect these fees by themselves, they hire a third party to do it for them - which is the above mentioned organisation.
(Most musicians don't even own the rights on their own music, because it's property of the label they're publishing with. But that's a different story )
The problem with streaming is, that potentially you have a worldwide audience, which - in the beginning of streaming - made such licences unaffordable and a lot of hobby radio stations had to close.
Meanwhile, the prices are much more reasonable as far as I know.
You'll find more information and pricing on the ASCAP-Website: https://www.ascap.com/music-users (I'm not a member nor in any way related to them - just happen to know a bit about this because I was a member of a community radio in Switzerland and we had to deal with all this).
Whether you pay the licence fees is up to you. Chances are, that no one who doesn't know the Streaming URL will be able to listen to it, so the music will only be played if one's in SL and at your club.
HOWEVER: If they DO find out, there might be a lawsuit for copyright infringement waiting for you, plus a ridiculous fee for unpaid licence fees.
That being said: I think you're one of the few here even thinking about this. I highly doubt that anyone who has an SL-club pays these fees. Last time I checked the Stream-Rentals, I don't know whether there was even a warning about copyright issues.
So, again - your choice :)))