I'm just looking at this now myself
If you have a JAR PPL you can hour build in the US under the visa waiver program. You are issued a restricted FAA licence based on your JAR licence valid for a maximum of 120 days. You must verify your licence which, as you say, takes 90 days. After a Flight Review (BFR) you are permitted to exercise the privileges of your FAA PPL.
You could, if time is against you, sit the FAA PPL theory test and flight test and receive an unrestricted FAA PPL but this requires at least an M1 visa and would obviously cost more. Visa waiting times are listed on the government websites.