An Internal Affairs cop was found shot in the head, in an apparent suicide supposedly, by her abusive cop husband who was recently forced out of the department for vague misconduct. There was an audio recording on her iphone, seemingly triggered by accident by whoever was holding the phone. The recording began three seconds after her last text message, it ended with her being dead, and it did not include a gunshot. So either she shot herself in the three seconds after sending the text message, and accidentally started the recording too late to capture the gunshot, or her husband shot her and sent the text message to cover it up and say she shot herself while he was at the store.

There are additional circumstances which cast suspicion on the husband. When he gets home from the store and supposedly finds her shot in bed, it sounds like he rummages around the crime scene for 50 seconds and turns on his cellphone flashlight, rather than saying anything out loud or calling 911 like he is surprised by a medical emergency. Her last text message, saying what to buy at the store and saying something to the effect of "goodbye world", was sent before her husband walked out the front door, while she was supposedly locked in her bedroom sleeping. She was supposedly an emotional and suicidal person.

Her husband was arrested for first degree murder, based on the idea she could not have shot herself in the three seconds after her text message and when the accidental audio recording began, combined with the other suspicious or at least supporting circumstances. This seems to be a problem where you can use Bayes Theorem to compare the likelihood of suicide or murder absent the three second problem, with the observation that the gunshot took place before the recording began. So if you think there is a 75% chance she killed herself when she is found dead without considering the three second evidence, and there is a 1 in 10 chance she would kill herself in the 3 seconds between message and recording if she killed herself, then there is a 77% chance her husband killed her.

Assume we cannot make any mechanical determinations suggesting who would be more likely to accidentally trigger the recording, a person who sent a message after shooting her, or a person who sent a message then immediately shot herself. Suppose we are totally lost as to trying to decide if it makes sense she would send the "goodbye world" type message and shoot herself in three seconds. How can we treat the chance she shot herself in exactly that three seconds, relative to the possibility her suspicious husband shot her?

Yeah, if you can find statistics on such crimes for the prior about the likelihood of suicide hypothesis and murder hypothesis, and then update them by the probability of each evidence given the hypothesis… It’s still a very subjective analysis, not very data-based.

wouldn’t analysis of the gunshot residue better show who shot?

I'm a little stuck on her text message saying what to buy at the store and then segueing into "goodbye world."

"Honey, don't forget to pick up a roast for when the Petersons are over on Friday. Also, I can't take it any more and I'm ending it all. Love you! Oh -- some of those pearl onions, too, if they have them."

People with depression often show sudden shifts in behaviour. They may seem unpredictable and contradictory sometimes.
OTOH, the outwardly "perfect" husband running errands for her could be a cover for some dark dynamics. Or it could itself be toxic: making the partner feel worse, dependent or a burden. He shot her? She shot herself? He pushed her to do it by some psychological manipulations? Difficult case, but if there was a shot, I'd think that forensic evidence would solve it.

I would investigate if the husband was a "Columbo" fan:

"A Deadly State of Mind" (Season 4, Episode 6): A psychiatrist kills the husband of his mistress and convinces her to lie to the police. Later, he kills her by making it look like she committed suicide.
"Forgotten Lady" (Season 5, Episode 1): A former movie star kills her husband and stages his death to look like a suicide
"Agenda for Murder" (Season 9, E3) A lawyer named Oscar Finch kills Frank Staplin to prevent him from revealing damaging information. Finch makes the murder look like a suicide, but Columbo notices inconsistencies that lead him to suspect foul play.