Hi Trini,
If I understood well, if not waiting for the temporary certificate, the only way to be sure that I can log P1 hour is flying with an instructor (JAA or FAA) or at least have a paper sign by one, but having restriction to take pax.
If you haven't got an FAA PPL (restricted or unrestricted) and you fly with an instructor then you will be PUT and will not be able to log P1.
However if you obtain an FAA Medical ask for a student certificate at the same time and you will then be able to fly solo on student pilot privilages.
Be warned though that if you are signed off by an instructor then even though you have an ICAO PPL you will still be classed as a student in terms of the FAA system and you will be very limited in what you can do, where you can go etc. Class B will be out of bounds unless you have a special sign off etc. Some Class B fields will be totally out of bounds anyway. Also iirc SVFR isn't allowed on a student certificate, just PPLs and above.
The instructor has to cover him/herself and will not allow you to fly unless it is crystal clear conditions etc... ie you will be very limited. This is due to insurance requirements etc...
The school themselves wil probably have very strict weather limits for student pilots for insurance requirements. Ie whereas if you had an FAA PPL issued you'd be able to fly up to say 17 knots crosswind in a PA28 they will may limit you to 10 knots crosswind as you'll be effectively flying as a student on a student certificate and instructor signoffs.
You'd be better off getting a full FAA PPL if you don't have time to get an FAA PPL under 61.75 (based on your ICAO PPL).
To obtain a full FAA PPL you have to ensure you have 40 hours flight time including the 10 hours solo, 20 housr dual, 5 hours night etc etc... then it is a question of 1 written exam and a few hours training for ground reference manouvers such as s-turns, turns around a point, lazy eights (that maybe only commercial)... anyway you get the idea.
Once proficient with these (they are easy and will take only a couple of hours to learn) you can get your instructor to sign you off for the FAA PPL Checkride. The FAA PPL Checkride will involve an oral portion where the examiner asks you plenty of questions about the aircraft / airspace / FARs / Weather etc etc... this will take anywhere between an hour or two. Then its a quick flight to ensure you can fly and upon successful completion the examiner will issue you your FAA PPL there and then (a temporary version with the real one in the post).
Obtaining a full FAA PPL will only take a few days after which you'll be able to fly to PPL privilages without the student restrictions.
Best wishes,
Charlie Zulu