I think I found him. There was some misleading info along the way - his birth date had been mistranscribed when he moved. This is what I did:
Working backwards from when he lived in Pitholm with first wife, all sources accessed via Arkiv Digital
Per Persson Berglund marriage to Anna Fredrika Marklund 3 Nov 1836 (3/11 1836) Living in Strömåhs but native to Piteå Stad
Piteå stadsförsamling C:2 (1823-1841) Bild 119 (AID: v138719.b119, NAD: SE/HLA/1010155)
Per Persson (Berglund added here) living in Stjärnan 59-60 Piteå
born 30 Sep 1812 (30/9 1812); moved in from Pite landsförsamling 1834, moved out to Strömnäset
Piteå stadsförsamling AI:3a (1832-1841) Bild 213 / sid 204 (AID: v138686.b213.s204, NAD: SE/HLA/1010155)
Per Persson (no Berglund) moved out as #62 to Piteå stad. 2 Nov 1834 (2/11 1834) Born 30/3 1812 (so maybe born in March?)
from p. 204 (hidden in the clutter at the bottom)
Piteå landsförsamling B:1 utflyttningsbok (1833-1860) Bild 77 / sid 73 (AID: v138636.b77.s73, NAD: SE/HLA/1010154)
So maybe this is him:
30 Mar 1812
Pehr born to bonden (farmer) Pehr Jonsson in Håkansön and its wife Christina Eriksdotter
Piteå landsförsamling CI:7 (1804-1815) Bild 130 / sid 250 (AID: v138647.b130.s250, NAD: SE/HLA/1010154)
Working forward from there:
Håkansön #16 (1807-1817)
bonden Pehr Jonsson 1779
hustrun Christina Eriksdotter 1777
Pehr born 30/3 1812 is #5 of 7 children
Piteå landsförsamling AI:4e (1807-1817) Bild 84 / sid 72 (AID: v138579.b84.s72, NAD: SE/HLA/1010154)
Håkansön #16 (1817-1825)
Pehr still in the household of his parents
Piteå landsförsamling AI:5a (1817-1825) Bild 73 / sid 63 (AID: v138580.b73.s63, NAD: SE/HLA/1010154)
Håkansön #16 (1826-1833)
Pehr moves out
Piteå landsförsamling AI:6a (1826-1833) Bild 195 / sid 191 (AID: v138585.b195.s191, NAD: SE/HLA/1010154)
Pehr Pehrsson b. 30/3 1812 goes to work at Håkansön #13
Piteå landsförsamling AI:6a (1826-1833) Bild 191 / sid 187 (AID: v138585.b191.s187, NAD: SE/HLA/1010154)
Pehr Pehrsson 30/3 1812 dräng at Håkansön #13 - moves to Pite stad 2/11 1834 Piteå landsförsamling AI:7a (1835-1843) Bild 211 / sid 204 (AID: v138587.b211.s204, NAD: SE/HLA/1010154)
So this connects with the moving-out book; I'm convinced it's him.