Before the pandemic, Air Tahiti offered lounge access at LAX (and other airports), though I don't know which one they contracted with. Someone here probably has prior experience to share. Their web site says currently only CDG and PPT offer this at the moment, though maybe that will change by the time you fly. Ask at the CDG check in what the lounge access is on your layover at LAX.
https://www.airtahitinui.com/us-en/airports-lounges
Yea, TN offers (at least pre-covid) lounge access at LAX but like most contract lounge situations this is only for departures, not arrivals.