I don't know if holding a JAR-ATPL will allow you to get a "reciprocal" FAA PPL - certainly my JAR-PPL was readily exchanged for an FAA-PPL although starting the paper trail takes a couple of months before you can present yourself at an FAA FSDO of your choice to fill out a few bits of paper and get a temp license.
Searching this forum will tell how that happens.
As I say dunno if you can do ATPL - PPL you may find the answer on the FAA website or by Emailing them in Oaklhoma City (address is also on their website).
It may be that Socal App who lurks round can advise.
After that, I guess it is just a matter of doing the FR (which in my case was abot 3 hours on the ground and an hour in the air) and pursuading somebody that you are safe to fly their 172 or whatever.