I recently flew US to New Zealand on a Delta ticket in business class, both Trans-Pac flights -- LAX-BNE and SYD-LAX were on Delta coded VA flights. VA uses the Emirates lounges at LAX TBIT for the BNE flight and the Etihad lounge in SYD international terminal for its LAX flights. (VA has a complicated set of lounge permissions across the map -- https://www.virginaustralia.com/us/e...lounge-access/)
But since I was DL ticketed (and I'm a Delta Diamond/ST E+) I tried to get access to the ST lounge at LAX-TBIT. I was told nope, use the lounge for the operating carrier -- so I went up a floor and used the Emirates lounge.
On my return in SYD, my connecting flight from CHC arrived a good 2 hours before the Etihad lounge opened so I walked over to see if by chance the ST lounge would admit me -- and they did. Not sure whether it was my VA boarding pass (which had my DL FF number and status on it, but the VA flight number) or my DL Diamond Medallion card with the ST E+ logo.
Then, because the flight departure was delayed I got a chance to check out the Etihad lounge as well.
All worked out for me.
tom

There is no Skyteam lounge in LAX TBIT. It’s a KE lounge used by all Skyteam airlines departing from TBIT but DL does not use this terminal. In SYD, this is a global Skyteam lounge.