No, I do not have access to early Jackson Co, MI, obituaries. But I took a look at the Heritagequest Census records and also LDS familysearch.
Heritagequest incorrectly transcribed the birth state for Caleb. He was born in New York, not New Jersey. This is confirmed by the 1850 census record. In the old handwriting "NY" looks a little like "NJ." So the Census shows Caleb T. Briggs as being born 1809/10 in New York.
The familysearch main database lists a Caleb Tracy Briggs, born 2 May 1810 Warwick, Orange Co, NY, who died 15 Nov 1863 in Jackson Co, MI, wife Elizabeth Ursula Morrill. In heritagequest 1810 census records only one Briggs is shown for Warwick, Orange, NY. He is John Briggs, and he is listed as having two young sons under age 10. So John Briggs is most likely Caleb's father.
Hope this helps. I suggest you try Orange County, New York sources to get more. Perhaps a genealogical society or library there can provide more information about John Briggs.