How do you help someone with severe PTSD get treatment if he doesn't want anything to do with the VA?

Sounds like getting treatment by a non-government PTSD organization would be best. There are many organizations besides the VA that help veterans address PTSD. They tend to be regional. For example, in Arizona where you are, there is another organization called RallyPoint (no relation to this site) that helps. The local VFW, AL, or DAV can also probably help with recommended service providers.

If he's still not willing to talk to anyone, then I recommend you reach out to those organizations on his behalf and try to get them to come and meet with him on a more casual basis. Don't set it up as a "PTSD session", but just "hey, I've got a friend you should meet." Once he realizes that there are people who are willing to help and that they are not enemies, he may be more open.

http://www.rallypointaz.org/

Vet Center is another org that is not VA run. You should be able to call Military one source and get PTSD contacts not in VA. Does he trust VFW, DAV, IAVA, wounded warrior or American Legion?

These groups have VSO who may assist, at least get peers to speak with.

One thing I would say is while it is commendable you are helping out an individual, there are craploads of people who have not even served, or have served and never deployed who use PTSD as an excuse for acting out. That he refuses help from the VA is a red flag for his claims. Although there is a lot of red tape at times it is completely insane that someone would turn down medical care and a pension if they are a legit veteran. I would suggest sending for his military records via SF 180. If you can get him to fill out the paperwork himself, If not you can still get a redacted FOIA form that will give you info on his service. You may want o contact a VSO in you area and see if they can help.