I've had a similar problem where as I purchased a ticket for a friend 3rd party and was told I must go to the Copa office to sign a form for my credit card. I understand why, but it was not practicle. I eventually found someone to put a note in my PNR that I would visit the office in LIM when I was there to take care of it. With some persistance you should work it out. This policy is easily enforced in Latin America where there are Copa offices everywhere and the people are used to picking up their tickets. But here in USA offices are in 5 cities .. they have to make an exception (at least they should).

Be prepared to send the voucher Fed Ex to LAX office with copies of your passport, etc. That may be an alternative, but I would not depend on the employees based out of Panama to accept that. You may have to call the LAX office directly. No guarantee this will work, but it's probably the best alternative .. or give the voucher away if it's transferrable.