One thing I comfort myself with (please note that I am not speaking for the Church, but I don't think she would officially deny the possibility either) ...
Jesus descended into Hades and preached to the souls there. They had the opportunity after death to "be saved". In fact we believe that Hades was emptied, I think. Of course, they were in a position of already having experienced death and time after that, and then Jesus shows up and preaches to them, so disbelief is probably an odd response.
But those who died before Jesus died had that opportunity. While I don't think we can say definitively that God offers the same chance to everyone, it is fitting of His character to do so. And we know that He desires all to be saved - He's not looking for excuses to condemn people on a technicality.
And when talking about blasphemy of the Holy Spirit, Jesus makes the curious statement that this sin alone of all sins will not be forgiven "either in this life or in the next". So the implication is that sin can be forgiven in the next life? As long as one doesn't blaspheme the Holy Spirit. Which is a matter of debate, but from what I've read, the Fathers seem to mostly say this means to recognize God as God and yet still reject Him (for various reasons).
Many people today reject the Gospel because the Gospel they are presented with is a wrong Gospel. God is mis-represented as someone who is only interested in "Thou shalt nots" or perhaps as a bloodthirsty tyrant who delights in eternally punishing people just for not being able to convince themselves of a particular fact (which may or may not be actually related to the true Gospel), or even someone who creates some (most?) people for the very purpose of condemning them just because it somehow boosts His own glory/ego (God forbid!). What they reject isn't really God Himself. So - maybe - they have an opportunity to respond too God Himself when they DO meet Him after their death? Or perhaps in that case they are judged by their own lives (which may well be another way of saying this defines their response to the true God when they meet Him) ... as the Scriptures mention in Romans 2.
We don't know that anything is fixed until after the judgement. We can pray for our loved ones. God is not bound by time, and prayers made out of love is how we are supposed to act. We have no teaching of how God will use those prayers, but we don't believe prayers made in love are wasted. So pray. And trust God.
(I can't bear the thought of people suffering in eternity either - that's the proper response of love anyway and how we should feel.)
You might like to read about St. Varus. I'll look for a link and paste it in.
ETA this one contains most of what I was looking for. I'm surprised how many google hits didn't include the info -
Icon of the Holy Martyr Varus (1VA01)