I would recommend you seriously reconsider appealing it. Just use that experience and move on to the next PD/SO and start over.
Most cops do not get hired their first time through the door. Also you can flunk a psych for one agency and pass it for another. They all generally have their own idea of the type of personality they are looking for to be their ideal cop.
Appealing the decision will only black list you from that department and possibly others. Call them up and ask if they can give you any suggestions as to how to do better the next time. Try to call the doctor and see if he can give you any suggestions. Finally find out what the repeat time is for LASD and then reapply. Let them know you are a big enough man to accept criticism, learn from it, change and try again.
Just my thoughts after 25 years in the business and managing Personnel as one of my responsibilities.