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I have no opinion about the health of the dog. However if it was not in full health, then there are only the following possibilities:

The brown dog was carefully brought by someone to the spot where it was wrongly killed by the LEO.
The brown dog got there on its own, slowly limping and clearly in pain. The LEO patiently waited for it to come closer, and then killed it anyway, just because. The other dog, of course, slowed down to the speed of the sick dog, as all dogs always do because dogs are not competitive at all.
The brown dog was capable of running. We cannot know how the dog felt; but the rate of approach prompted the LEO to fire at the dog that he considered to be the most dangerous.
Both dogs were walking to the LEO. But the LEO decided to shoot one for no good reason at all, even though this would definitely cause problems at work and may frighten his elderly mother who was just a few yards away inside the house.

If the neighbor has issues with performance of a city employee, the proper and socially appropriate way to do it is by talking to his supervisor. Perhaps then she would learn that the LEO made special arrangements to care for his elderly mother. Or, perhaps, such a radical idea had never occurred to the neighbor?

This reminds me of an old joke:

As the orchestra is playing, the conductor receives a note: "One of your musicians is lazy: he plays his instrument only when you look at him."

On some forums it is fashionable to accuse police of all sins. However I think it is far better to be fair, and judge people for what they have actually done, no matter who they work for. In this case the LEO acted as any citizen could act. It's not like he kicked the door in, with a search warrant in hand, and shot every dog in every locked room.

The motives of a neighbor who accuses anyone of caring for his mother are extremely suspicious. For example, she claims:

“He’s over here at his Momma’s house drinking coffee, eating breakfast I guess

"His Momma" suffers "from a medical condition called cerebellar ataxia, which causes problems with balance, gait, extremity and eye movements, [...] Mrs. Azwell, who is four-feet, 10 inches tall and weighs around 100 pounds, is a “fall alert” patient" -- yes, of course she would be cooking breakfast for her son and making him coffee - and not the other way around, as any reasonable person would presume. There is absolutely no reason whatsoever that a frail woman would need assistance to get out of the bed and start her day, in opinion of that neighbor. She isn't painting a very good picture of herself, is she?