There is no formal requirement for 'supervised cross-countries with an instructor and an additional exam'. Neither may NAAs gold-plate Aircrew Regulation requirements to suit themselves.
However, it's possible that a flying club might have its own requirements before it is happy to lease an aeroplane?
Incidentally, IAOPA has proposed that the '10 hour' clause, which for some reason doesn't apply to the LAPL(H), should be amended to allow novice LAPL pilots to carry passengers without the 10 hour PIC restriction, provided that such flights are supervised by an Instructor. I put this to the last FCL Implementation Forum and it received unanimous support, so it is now up to EASA to make the appropriate change. However, they move with the speed of a handicapped glacier, so don't expect anything soon...