I think the missing part of the equation is time. The OP is wondering why gas in a tank at say 3000PSI at room temp (say 70F) ends up in another tank at 1500 PSI, but is now warmer, say 90F. I think he's wondering where the heat energy is produced. It 'should' be such that the net cooling and heating is zero, or less than zero, rise due to the net loss in pressure.
And actually, if you were able to measure the actual temp of each air molecule and track it through time, you'd probably find that was the case. But simply feeling the tank, you're feeling radiant heating/cooling from contact with the air, and the metal in the tank absorbs and retains this heat relatively slowly. And, it's probably a different rate of radiance depending on the ambient temperature and maybe even the properties of the metal, I really don't know. But certainly if tank one cools off as air goes from 3000 PSI to near ambient, that 'could' be a measurable amount of heat release, then tank two heats as the air goes from near ambient to 1500 PSI, and that 'could' be measurable, and if so, the net heat release would exceed the net heat increase. And if you could somehow measure these amounts and display real-time numbers representing them, you'd see the delay in radiance pass from one tank to the next, and to the ambient temp, eventually equalizing. And if you could measure actual total heat loss/gain, you'd find a net loss, but only after the radiant effect on the tank metal had passed. At least that's my guess!