Per the conditions of membership, you can access the SC when flying another airline as long as you can access the appropriate concourse, terminal, etc. where the SC is located. I'm not that familiar with FLL (been many years since I was last there) but if your BP won't get you to the right concourse, then your options are: get a gate pass, sweet talk your way through TSA (which will sometimes work if they know you're going to a lounge), buy a fully refundable ticket that will allow you to get onto the concourse, use the SC and then refund the ticket later, or spend your time at the gate.