Yes, I do believe that the purification that takes place during the Dark Nights, if not done in this life, would have to take place in Purgatory.
And I have complained a bit about it before, that the Western emphasis about Purgatory has always been on satisfaction for sins committed, but I believe that perfection/sanctification is also required. So, for instance, even if a person were free of all temporal debt due to sin (e.g. by receiving a plenary indulgence), this does not mean that the soul is prepared to enter into Heaven and the Beatific Vision. That requires perfection and sanctification. Of course, God can if He will accomplish that in an instant.
In other words, I believe that BOTH satisfaction and sanctification are required. In a sense, the two are related, and perhaps it's just a matter of emphasis between the East and the West.
I've also wondered about the following scenario, where you have an otherwise sinful Catholic, with many defects, who is martyred (almost passively). While the martyrdom most certainly remits the temporal punishment (satisfaction) due to sin, that soul would still be in need of some serious purification and sanctification in order to be fit for the Beatific Vision.
Then what if you have an unrepentant mortal sinner who suffers martyrdom passively. Let's say he's just walking in to Sunday Mass and some Muslim sniper shoots him in the back of the head on account of his being a Catholic. He didn't know what hit him, and he never had the opportunity to repent of his mortal sin. What then?
There's a lot of strangeness with "Baptism of Blood" doctrine that seems unclear and pretty foggy. Generally the Church Fathers attribute remission of sin to Our Lord's teaching that "no greater love" can one have than laying one's life down for the faith, i.e. that there's an ex opere operantis aspect to it, but later it was transformed into an ex opere operato effect, where even passive martyrdom without any heroic virtue involved would have the same results. I think that something went off the rails somewhere.