Any suggestions on the best lounge at LAX if we are flying out of TBIT?

We are flying with China Airlines in J from LAX to TPE and then we might be connecting to another city in Y.
If I understand correctly this notice on the Skyteam website, would only apply to us at TPE and we can still access the lounge at LAX, right?

“For connecting customers, both the domestic and international flights must be operated by a SkyTeam member airline and all segments need to be booked in First or Business Class.”
st or Business Class boa