I may be able to help, to some extent.
While I don’t know for sure if the purification process in Purgatory is temporal punishment, I do know that we receive temporal punishments right here on earth.
For example, if somebody commits a crime in this life, their temporal punishment (which I define as a specific tangible punishment) could be time in jail or a long prison term. Temporal punishment is usually a specific consequence that the offender has to experience and endure.
If somebody embezzles funds, they can be forced to pay back a specific amount, or make other specific and tangible restitutions.
I think what’s confusing you is the definition of temporal punishment. It’s something specific and tangible to the person receiving it.
The purification process in Purgatory may involve some of that, or it may be a generalized, overall purification, involving the totality of the human soul.
The truth is, we really don’t know what we shall experience in Purgatory, only that we shall be thoroughly cleansed and purified of whatever small imperfections remain.
And, because we are no longer inside of our physical bodies when we die, all of that shall occur supernaturally or completely spiritually. So, the question actually is, can we still experience specific, tangible punishments (or consequences) after we are no longer inside of our bodies? Maybe yes, maybe no, or maybe it will be a different kind of experience, altogether.
That’s my take on it – I may or may not be entirely correct. None of us will know until we get there.