Are the sources also wireless? Because if not, all you'd need is a 3(+) channel mixer. Route your inputs to the mixer channels. Connect a BT transmitter to your booth or master mixer output. Then just use the mute buttons on the mixer to decide what to listen to.
As far as the mic thing goes: I don't think what you're describing is going to work since BT audio drivers for headsets are a little too finicky for that application. Best bet is to just buy a tabletop mic with a 6.3 jack. Depending, again, on how you want to use it. Because a mix would go into its own mixer channel. You can't really send input back to another input.