Saying $CO_2$ liquefies at -56 Celsius is an oversimplification. At that temperature the vapour pressure of liquid $CO_2$ is about 5 atmospheres, so your liquid would be in equilibrium with about 5 atm of gaseous $CO_2$ together with however much $O_2$ and He. If you want the proportion of $CO_2$ in the gas part to be (say) less than 0.1% (2.5 times that in Earth's air) you would need about 5000 atmospheres of oxygen and helium, making the compressor and tank less than simple. The same link gives data for other temperatures, for instance at -120C the vapour pressure is 0.013 atm, so compressing your air to 13 atmospheres and cooling to -120C might be a more reasonable target
A further issue is the cooling. Compressing the gas will heat it considerably. That heat could, perhaps be lost by conduction back to the rest of the spaceship (although that really just shifts the problem). After that, if you don't want to put in still more energy for refrigeration, the best you can do is put it in a black container with lots of radiator fins and put it on the shady side of the spaceship to cool by radiation. This is not especially effective, especially as the container gets colder. For instance cooling by radiation at 200K (-73C) assuming no incoming energy at all, loses about 80 $W/m^2$ (using the formulae from here). The specific heat capacity of helium is around 3000J per kilogram Kelvin (at constant volume), so it's going to take a while to cool down much gas.