Cold crash amount of CO2 sucked into fermenter

I plan on ussing a bag/balloon filled with CO2 attached to the fermenter when cold crashing. I want to calculate how big this bag needs to be.

Assumptions:

• Fermenter Volume: 30l (23l beer)
• Room Temp: 20° Celsius
• Beer Temp before Cold Crash: 20° Celsius
• Beer Temp after Cold Crash: 3° Celsius

How can I calculate the volume CO2 that is sucked into the fermenter?

My approach was to look at the Density of CO2 at the different temperatures:
1.815 g/l at 20° C
1.923 g/l at 3° C
So at 20°C there are 54,5g CO2 inside the fermenter at 3°C there should be 57,7g CO2 in the fermenter, the difference would be 1.8l CO2 at 20°C. I feel that I'm missing something, probably something about the CO2 in the beer.