Book a week or two holiday in Florida and with a flight school that knows the system for getting your TSA clearance, pretty straight forward, I used Tailwheelsetc. near Orlando, as I was going on holiday and "not" just for primary flight training I didn't need to jump through any hoops regarding a visa.
While I was there a guy from Ireland came over to get his FAA ticket, he knocked it out in a week in fact he walked in on Monday morning and took his checkride on Friday afternoon and was on a plane back home on Sunday.
As you are already a pilot, it will be mainly learning the US airspace system, which to be honest is not difficult.