How does gas equilibration occur in the alveolus, numerically?
There is no known mechanism for exchange of $\ce{O2}$ and $\ce{CO2}$ in a one-for-one fashion. Their transport across the alveolar membrane takes place by diffusion alone: both of them attain their equilibria with blood independently.
Note that it is not necessary for inhaled and exhaled tidal volumes to be the same. In fact, under physiologic conditions, the exhaled tidal volume is slightly less than inhaled tidal volume. This is because humans normally produce slightly less $\ce{CO2}$ than they consume $\ce{O2}$. In other words, the respiratory quotient is less than $1$.
Thus, we see that the assumption of preservation of alveolar volume and pressure is not valid. Apart from inspiration and expiration, the diffusion of gases to and from blood also affects alveolar volume.

Reference and further reading:
Boron WF, Boulpaep EL, editors. Medical physiology. 3rd ed (international edition). Philadelphia: Elsevier; c2017. 1297 p. (I found this textbook an excellent resource for learning physiology in general and respiratory physiology in particular.)