Your only reasonable options from transportation between LGA and Cape Liberty are rideshare (Uber or Lyft) or a car service.
Public transit between the two is impractical, requiring 5 or 5 different transportation mechanisms combining bus, multiple trains, and a taxi or uber ride at the end.
It seems fairly obvious from reading the pp2f website that they only provide local shuttle service to and from their parking facilities in the immediate vicinity of each of the three airports, not service between airports.