Forensic firearms experts would test Baxter's hands and wrists and shirt sleeves, etc. for gunshot residue. There will be a distinctive pattern of residue and that pattern would have specific scientific characteristics. The pattern ought to match up with other evidence, including how the weapon was held when it was discharged.
My instincts tell me the wife and the victim's family ought to be requesting a second round of forensic testing to verify and confirm initial conclusions.
If the man was Catholic, or was educated in Church schools or universities, I must resist a finding of 'suicide'.