My name is XXXXX XXXXX I'd be happy to answer your questions today. I'm sorry to hear that this happened.
Paypal unfortunately will not release that information without a subpeona. To get a subpeona, you would need to have a pending court case - which is a Catch 22, because you don't know who to sue without the subpeona. What you would have to do is file a lawsuit against "John Doe," then subpeona the information, trace the IP address, and then, if you can figure out who it was, amend the Complaint with the person's real name and address (then serve them with notice of the suit).
Considering that you are in another country, the best thing to do is to file a police report with the local police in the place where you live when not deployed. You may also want to contact the FBI, as they deal with internet fraud. Police can get IP address information from Paypal.
Reporting the theft also will give you a paper trail, which can help you if any questions arise related to your secuirty clearance.
If you have not already done so, instead of just added security measures with TransUnion, Equifax, and Experian, you may want to consider a total credit freeze, so that any requests for credit will be automatically rejected. The freeze can be removed after you get this whole thing resolved.
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