What FAA license does the EASA LAPL convert to?
Unfortunately, an LAPL license isn't ICAO compliant. This is from the UK CAA:


  Licences which do comply are known as ICAO licences and those which
  do not are known as non-ICAO licences. Non-ICAO licences are not fully
  recognised internationally, and therefore are only valid for use
  within certain states or Europe; for example, the LAPL is only valid
  in Europe.


That means that the FAA won't recognize it and you can't use it to get an FAA PPL under 14 CFR 61.75, because that requires you to hold an ICAO compliant private pilot's license. As far as I know, there's no other way to convert a foreign license to a US one (a sport or recreational pilot license would be the closest US equivalent).