Thanks all for your input. I'll type up a report here on my experience in getting into the AC without being on a flight.
As I said before, I called the Executive Center and the agent told me that they'll get in contact with the LAX AC although all I needed to do was to go up to the TA counter to get in. I cited the rules that it wasn't the case so promised to call me back later. I never did receive the call back.
Went to my home airport and went up to the counter. Ticket agent was unhelpful; she initially gave me the cold shoulder saying she wasn't supposed to do that and that I was supposed to contact the airport's AC directly. When I cited that I had no direct way of reaching the AC due to the change made in the 1-800 phone menu options, she gave me a huge sigh and called the AC.
After a few minutes on the phone with someone at the AC, she said I was lucky that I was on the list for that day and printed me a fake boarding pass from Sabre Field to Headquarters. But before handing me the boarding pass, she said it wasn't her job to give these out - that it was the AC's responsibility to come down and give it to me personally. What? How on earth does the AC know when I'm supposed to come and how am I supposed to call them to let them know I'm here when there's no way to get in contact with them? I was already scrapped for my meeting time so I just said thank you and went on my way.
Now for those of you who know LAX, it's one of the airports where the TSA actually scans the BPs before going through the screening. When the TSA took my BP and scanned it, it gave a red beep. The TSA agent was confused so he looked at my BP and said that this wasn't a valid BP because the departure city said Sabre Field instead of LAX. I politely said that this was a BP issued to get into the AC and showed him my membership card. He called over his supervisor and after a few minutes he let me through.
Ugh, so much hassle just to get into the AC on a day I'm not flying. Next time, I'll insist on having a meeting landside!