I venture to say Laurentius Beaver was Peter's Uncle, Peter is listed as an heir in Laurentius's will, not a son. In researching baptism records of Laurentius children, there is not one for Peter. (Not one for Nicholas but Nicholas was called son in will). In the book "Eighteenth Century Emigrants From the Northern Alsace to American" by Annette Kunselman Burgert, Page 64 doesn't list Peter as a son of Laurentius. This book list American records and British records. Very interesting and somewhat confusing.
A quick look-around of web trees finds many that do not know who Peter Paul's parents are, but some who do, and they attach him to Laurentius.
If there is indication that Peter Paul is not Laurentius' son, then I suggest we move him away. The main question is: to whom? There are several possibilities; need to take some time to figure out which are plausible. The fact that he went to North Caroline is helpful, since this would allow him to be overlooked more readily in Pennsylvania.