I live in CEB most of the time, and travel to LAS several times per year, and I've been buying my tickets over the phone from the local CX offices for 2-3 years without any trouble. Usually the CEB office but sometimes MNL if the CEB line is busy. Never had to show up in person, or email an authorization form, or anything like that. Just give them the CC# over the phone.

And yup, all of the travel starts here. All of my trips include AA portions LAX-LAS. Bought with a foreign credit card.

Maybe they've modernized their processes in the past few years? IIRC I used to get my tickets on the CX website until around 2011 or 2012. Around then there was a website upgrade that broke CEB-LAS flight searches for premium tickets and I've been calling the local office ever since.