Experienced technical diver here (1000+ dives)
In principle, Triton could work IMO. I enjoyed some of the details about NASA estimates of oxygen use but I think you're on the wrong track analytically. At many points in your analysis you use oxygen and air (let's call this "tidal gas" because it could be air but it could also be some other combination of gasses) interchangeably. The difference is a big deal. Oxygen is a chemical needed by humans for respiration. Air is a roughly 1:4 mix of oxygen and nitrogen with a bunch of trace gasses.
A quick point about fish before we dive in. You're right that they're a poor analog but some of the details your provided were a bit off. Fish don't breathe at all. They do not extract air (defined as a mix of oxygen, nitrogen and trace gases) from water. Instead, they extract dissolved oxygen from water through gills. This is analogous to the way that our lungs extract oxygen from the air mixture but with some important physiological differences.
And a word about oxygen. Unless decompressing at a very shallow depth (<6m generally), divers never breathe pure oxygen. Below 6m, pure oxygen is neurotoxic.
Humans use a very small percentage of the oxygen we inhale. The rest is exhaled along with expired co2 and the nitrogen or other gasses in the "tidal gas". This means that only a tiny fraction of the tidal volume needs to be replaced with each breath. This is the basis of rebreather technology used by divers and astronauts today. A very small tank of oxygen provides many hours of breathable air by adding the utilized oxygen and removing co2.
For Triton to work, it needs to either extract dissolved oxygen across a semipermeable membrane or, as you pointed out in your discussion of submarines, utilize electricity or a chemical catalyst to extract oxygen directly from h2o. Since water is almost 80% oxygen by weight, electrolysis seems like a good bet. It's not terribly efficient from a power perspective but it is highly effective as many of us saw in high school science class.
However, your point about power requirements and miniaturization was wrong. This is done today in rebreathers which use a closed loop counter lung. Here's how it works:
When a diver exhales, her lung volume decreased and the air is pushed through a one way valve which leads to a chemical scrubber.
After the chemical scrubber removes the excess co2, an oxygen sensor detects the slight drop in o2 relative to the target set point for that depth and adds a tiny puff of o2 to reset the mix that now inflates the counter lung (basically a bag of breathing gas).
The amount of o2 required (in grams) is independent of depth and largely independent of effort on a per breath basis but more will be needed per unit time if the diver is exerting themselves and breathing more rapidly.
When the diver inhales, the counter lung empties the fresh mix into her lungs and is ready to start again.
Triton needs to be able to generate that puff of o2. This seems very feasible.
I suggest we keep an open mind and support these guys. Running out of gas at depth, sucks. I've rescued a few folks in that position and it is as flat out an emergency is in diving. If the Triton team can develop better alternatives, it would save lives and open the ocean greater exploration.