Status shouldn't really matter - but being OWE gets you to the F check in counters, that should have 'better' agents...CX should check in to all interline partners (unlike BA, who will only check through to oneworld airlines on separate tickets).

I've checked in (CX->CX)->LA->TA across three different tickets..took, a bit of time, but got my bags (among the first to pop out!) at LPB, after a SIN-LAX-LIM-LPB hop.

Might help to call the airlines to cross reference both the PNRs/ticket numbers..and keep the e-ticket numbers handy...