After some research and lots of luck, I basically stumbled on what is more than likely Johann’s birth record. On a whim, I just decided to start looking through records just to the north of the Plathe area (where Johann was married), since I didn’t get to those on my last search. I wasn’t expecting much since it’s not that close to Naugard or Minten. I started with the parish of Batzwitz (now called Baszewice) which also included a few other smaller parishes in the area. I jumped in a few pages to skip over the cover page, etc, and the first page I came to had a baptism entry for LAST. Though it wasn’t anything familiar, I did take it as a good sign. In all of my searching, I found that the surname isn’t very common. It was only in 1823 and Johann was listed to be born in 1825. I kept working through the pages, seeing more LAST entries.
Then, on the first page of the 1825 baptism records for the parish of Barkow, I spotted what looked like “Johann Wilhelm Gottlieb,” his full name from his marriage record. Unfortunately, it seems that Johann is listed as uneheliche sohn which basically translates to “illegitimate son.” It does list his mother as Dorothea Sophia LAST, but no father. I know that some church records sometimes have a “legitimacy” section that has listings for when they legitimize the children, so I’m hoping it’s there.