Your right, LOX or solid form of O2 space wise would be about the same. LOX would be pure and not require any burning. That makes a lot of sense.
It makes no sense at all: "LOX would be pure and not require any burning." What are you trying to say here? Burning combines oxygen with fuel to produce heat. Pure oxygen is more efficient than air as an oxidant (air contains 80% nitrogen), but the oxygen still has to be consumed in a fire to produce the required heat to drive an engine.
Sterling engines run on temperate differentials.
In fact, all heat engines run on temperature differentials, including the common ones like internal combustion engines, steam turbines in power plants, and jet engines. But always you have to put energy in to get power out.
I would think a Sterling engine could easily run on the temperature diff between LOX and the coldest seawater.
Again, no. This is the "free energy" trap. You cannot get energy for free. To get power out you need to put heat in, whether it by by burning fuels, or from nuclear fission, or from renewable sources (solar or wind, since wind energy comes from the heat of the sun). If you try to take the required heat energy from the sea it will simply heat up the LOX and boil it away, quickly destroying your cold reservoir.
If they were going to do this, they wouldn't use LOX (highly dangerous), they would use liquid nitrogen or dry ice. But they don't because there is not enough energy available to derive a useful power output. Combustion is the only way to get enough power to drive a useful engine (unless you use batteries).
But it appears they burn diesel fuel and use the LOX as the oxidizer. What happens to combustion byproducts? Do they use a CO2 absorbent? Or vent the exhaust into the ocean water? But if they vent, at say 100 feet underwater that would be around 3 atmospheres. So wouldn't they need a pump or compressor to pump the exhaust gasses into the seawater? And then wouldn't the bubble give the sub away if no the pump or compressor?
These are good questions. Possibly the sub absorbs the CO2 since venting it would give away the sub's location.