I'm confident that we've identified the Joseph Septimus Carr, father of Thomas Carr as Joseph Septimus Irvine, born 1813, son of Thomas & Mary Irvine (Tebay) and baptised in Lancaster.
Thomas Irvine died when Joseph was 2. His mother married John Carr and this seems to have been the origin of the middle name Carr which Joseph gave to some of his sons, and the name which his eldest son, Thomas adopted as his surname when he moved to Liverpool with Catherine.
Joseph married twice, both wives called Ann. He lived in Preston with 1st wife and in Blackpool area with 2nd wife (but he waited until Ann #1 died before he married Ann #2).
Thomas Irvine was born in Preston, son of Joseph's 1st marriage. 3 of Joseph's sons were named John.
The death of Joseph Septimus Irvine was registered in a sub-district adjacent to where his son, Thomas Irvine lived and far from Joseph's home in Blackpool.
The reason for Thomas changing his name to Carr seems to have been connected with Catherine already having a husband, Francis Johnson when she previously lived in Liverpool.