(heavy speculation) My belief is that a oneworld airline is required to give the same lounge access to OWS as it gives to J passengers -- this is demonstrated by AA letting elites into the Flagship Lounge when United does not let elites into the Polaris Lounge, and by AA and other oneworld airlines giving elites invitations to contract lounges. Therefore if LATAM uses the Sky Club for its J passengers, it may be obligated under oneworld lounge rules to provide access to the same lounge for OWS and OWE elites.
Given that Delta is the one operating the contract lounge, it is possible Delta will tell LATAM that business class and LATAM elites (and DL elites flying LA) are welcome but other oneworld elites are not. In this case, my guess would be that LATAM would say "forget the rules, we're leaving anyway, they probably won't sue over a frequent flyer thing" and shut out oneworld elites.
Even if LATAM does end up allowing oneworld elites into the Sky Club, it's going to be quite a sight trying to convince the dragon at the Sky Club "I'm an AA Platinum, that gets me Sky Club access flying LATAM for the next 9 months."