Neither AA nor TN is a Star Alliance carrier, so I'm not sure why you posted this in the Star Alliance forum.

In any event, you must always check in with the operating carrier (TN in your case). And it appears that TN Business Class passengers get access to the Star Alliance lounge in the Bradley International terminal at LAX:

https://us.airtahitinui.com/los-ange...al-airport-lax

("Full-Service Lounges")