'As deadly': doable. for one shot. The 'Texan' is marketed as 'the worlds most powerful production airgun' and is supposed to reach about 800J of muzzle energy. For comparison, a NATO 7,62×51 is listed by Wikipedia as having around 3000J of energy (NATO 5,56×45: 1500J), so while the airgun is somewhat lacking, it is already in the same ballpark. Airguns for research purposes often use hydrogen or helium as propellant gases, because their speed of sound is higher, enabling higher velocities for the bullet - also, higher temperature of the gas is good, which they achieve... with a real gun
But lets assume we have a working gun, the burst disk from the research guns has been replaced by a special valve (or has it? maybe your gun just has a big supply of them) and the gas is somehow heated (electric arc? would fit the theme: an air-electric rifle sounds awesome) and we now can get a projectile to competitive muzzle energies - how often will we be able to shoot? Ideally, the bullet gets pummeled by compressed gas all the way to the muzzle, meaning that the entire barrel needs to be filled with compressed gas. let's say that is a 1.28cm diameter barrel, and we assume a length of 50cm, that will be 50ml.
These are 50ml at high pressure, near the pressure of your storage, so let's just pretend that is the kind of storage volume you would need. That makes a 1l vessel capable of 20 shots, which is not realistic of course, because the vessel looses pressure every shot... The Airguns of yore (mid 1800s gun) had that same problem, loosing power with every consecutive shot. Todays composites allow the airtanks in airsoft to be at 30MPa while being quite portable, steel tanks are not really bigger, but just very unwieldy due to their weight.
Energy density is not very good for compressed gases - additionally, the way you get the energy out of there matters a lot, and 'violently pushing small object down a barrel as fast as possible' does not even begin to be an efficient way of getting your energy out there. Coaxing the energy out in a slow, multi-stage process, will yield 300 kJ from a 5l bottle a 20MPa, already discarding another 230kJ that are lost, or stay in the bottle which left filled to 1MPa... (300kJ was also about the energy you cited for your soldiers loadout) - 1l Paintball tanks at 30MPa will give you about 1000 shots at about 10J, so 10kJ are harvested, while the above example shows that for the same volume at only 20MPa you could get 60kJ
But maybe you should own and compensate the drawbacks of the airrifle and not make it about pure muzzle velocity (and also not about muzzle energy - kinetic energy scales with mass, but it also scales with velocity squared, so there is no way to keep up an energy arms race if you cannot score velocitywise) while playing to its strengths:
- Keep the gas-supply topped up via chemical reactions: just like cartridges, but slightly slower burning and thus completely silent
- keep muzzle velocity below the speed of sound for gun-noiselessness and reduced requirements regarding pressures and gas-types
- keep the barrel very thin to avoid losses
- gently shoot stuff that does not like being explosively propelled (seeker-bullets? bullets with little razor wings that swing out and help with range? bullets with some kind of tracker (for the big guns to target?) ?)
- rev up that rpm (no barrel overheating, if anything they might get frosty; no need to get rid of cartridges)