The first reason to believe is simply for the sake of hope.
In the parable of Lazarus and the rich man, Jesus talks about a world to come.
In that world, the dead rich man asked that Lazarus be sent back to warn his “brothers” of their impending doom. We know God does not hear “sinners”, but this prayer of the rich man apparently was heard. For Jesus did raise a Lazarus from the dead; the namesake of the character in Jesus’ own parable. Lazarus, therefore, presumably died twice or went to the world to come (in two states).
If Lazarus had gone to heaven, with God, how can God bear to be separated again and again to die? We’re told that no one can take a prize away from God once he has it.
I mean, even the Blessed Mary, when she appears in visions (which you aren’t required to believe in); is not to be understood as coming back to Earth to die a second time.
I’ve sometimes wondered, in a bit of a fantasy, what would happen if I had nannites from the show “Dr. Who.”; and a girl whom I’m sweet on, died. (There is one who’s alive, but I just pray for her and have nightmares about rejection if I saved her life after death.). eg: If I re-animated her body with nannites … before it corrupted, would her spirit come back? Would she hate me or be thankful?
For, if she was already with God … nothing could separate her from eternal happiness; so I could only raise her if she were in purgatory or hell. eg: like Jonas and the leviathan from the old testament. It’s not much different than CPR bringing a person back from death due to heart failure. The techonology is just incidental.
And don’t fear that I’m after immortality on Earth. Jesus will come a second time as the universe ends … and all who are destined to be with him, will be changed “at the sound of the trumpet.” I Corinthians 15:50-51.
So, I’d like to reverse your question, slightly, and ask you to look at it from another direction; “Purgatory”, is it only after we die? … or is it not, also, the removal of all stubble here on earth until the moment of death. For, that which is already here should not be surprising to find continuing on the day of Judgment to finish cleaning up even the very last thoughts of evil or ignorance that a person might have.
(eg: To paraphrase, I think I Corinthians 3 … something…)
“For no one can lay a foundation other than the one that’s already laid.
Whether anyone builds upon the foundation using gold, living stone, fire, wood or hay, it will all be brought to light on the Day of Judgment. For the man’s work will be tried [future tense] as if by a fire (Pyre/pyrgatory) and the fire will reveal the quality of the man’s work.”
The idea of fire is perhaps metaphorical, and I don’t visibly see it in everyone’s lives here on earth. Therefore, I think for some the fire must ignite after they leave Earth. “but the man himself [shall be] saved.”