^^^ This.
FAA students may log PIC when they are rated in the airplane.
So a PPL holder under Instrument instruction or a PPL holder under CPL instruction.
There are more differences.
Under FAA the Instructor is obligated to sign the students logbook for every flight in which instruction is given.
The is not done in Europe for instance which makes it harder to distinguish what exactly took place.
I’ve had many European trained students that lacked sufficient logbook evidence to meet certain FAA requirements.
If I trained initial European students I would show them how to log FAA PIC and EASA PIC in two seperate columns in their logbook.
So back to your case.
You need to find out what the Aviation Authorities require as far as logbook evidence and Training records to satisfy THEIR requirements.
It may very well be that they would not accept the FAA dual/PIC either.
You can’t “make” PIC hours by converting licenses.
Don’t worry about what could have been if you’d done your training elsewhere...since you didn’t.
Yes, you may have to fly some more.