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#7 above is 100% correct; AA has no lounge or lounge arrangements with 3rd parties so regardless of OW agreements or status there is no lounge to use when flying AA domestic (there is no AA international from this airport). I'm not sure how long your connection time is between BA and AA, but LAS airport is not the greatest for connections so leave plenty of time. Arrivals at T3 on BA can sometimes take over 90 minutes to clear immigration & customs (do you have Global Entry or are you a US Citizen, in which case you will be out quicker), and baggage delivery is notoriously slow at this station. Once you clear customs you'll need to change terminals landside and go through security again at Terminal 1, which depending on time of day can take over an hour due to all the tourists in this terminal (unless you have TSA Pre-check or are travelling AA First Class). I'm not trying to freak you out, but LAS is not really set-up as a connections airport (unless you are making a domestic to domestic connection on Southwest), so when you make a connection of this type it takes longer then it would at a hub-style airport).
Assuming you have lots of time and cannot use the Amex Platinum Lounge (which is hands-down the best lounge at LAS by far), you can always enjoy the slot casino and bar airside in terminal 1 (and if you are a smoker you can also smoke there).....