In addition to 61.109, 61.41 is helpful. You'll just need documentation of the training received. In the US, instructors sign off logbook entries. Not so common in some parts of Europe...
Also, if the primary purpose of your visit to the U.S. is flight training, you'll need a visa which can only be obtained when working with a SEVIS-certified Part 141 school.
As for the piggyback: in order to use the FAA certificate, you need a fully valid and current European license/type rating/medical. Obviously, the OP doesn't have that. You also need a current flight review with a U S CFI Per 61.56, but that's another matter.