rudestuff, I am guessing chicken will need a visa and thus a 141 school anyway. If not, part 61 all the way for sure, although there are more hours, they will be both cheaper and also pay as you go and flexible on timings/doing it fast if needs be. And they don't have to have EASA med to get 61.75 issued, you can use a FAA med, sensible to get a class 1 faa done asap. The verification letter will have the medical on it if valid, or just verify the ppl if no current medical.
I just have two columns in my logbook and log easa and faa separately. Best though is to log it the FAA way - meaning not just put PIC but have the instructor endorse each line as you go as they seem to require, then when totalling stuff you can claim pic for faa but there is no ambiguity that you are trying to cheat easa as it is clearly dual - if that makes sense.