I would think the exact opposite would happen actually.
If you squeeze the bottle then cap it, the liquid in the bottle at the top will be exerting pressure on the walls, trying to make it go back to its original shape so it can sink down and find it’s lowest level. This will create a volume of reduced pressure at the top, which will “suck out” the dissolved CO2. You’ll end up with a bottl that looks unsqueezed, but the “air” above the liquid will be all CO2.
I would guees you’d do better to pump air into the bottle somehow. With high pressure above the liquid, less CO2 will be able to fizz out of solution.
//But it was a long time ago damnit.