I can't offer any advice other than to write again to BA (cc. LAN). As you say, it is your BA status that's missing, so BA are your main point of contact (although your are LAN's customer, I guess).

If that fails, you could try a PM to Nicci, (username:BA Executive Club), but she is here as a backstop when usual channels of communication have failed, and I believe she gets very busy, so try writing again first.

My personal experience is that my BA/EMERALD/xxxxx always shows up, and LAN allow me to allocate seating etc.* (I usually do it on QF's website, as my "normal" LAN legs are after a SYD-SCL QF flight).

*not sure what other benefits I can have/miss as I don't take much luggage and I get J lounge access on a D class ticket anyway, and I don't recall seeing an F lounge on the LAN network.