Similar enough question not to start a new thread.

Do I have lounge options flying LAX to FCO then FCO BDS (Schengen)? (booked on DL)
Is there a lounge at BDS for my return to FCO that I can access (booked on ITA...amex reserve and DL platinum member.)

It looks to me that Piazza di Spagna wouldn't be accessible upon arrival. that you're routed through passageways to passport control rather than having the option to stick around at gates?
Is my only option to go for Hangar lounge if that is in the gate area I'm departing from?

I arrive at 6am after a long trek from LAX-JFK- FCO and have 4 hours wait for friends heading to Brindisi which doesn't seem to have a lounge outside of the airport lounge


Thanks in advance for any direction before Im mentally exhausted. this trip is 2 weeks after my return from a month in Europe. I am anticipating a bit of time-zone whiplash!