It is only available on Hard Difficulty!
You can fight it in the Shinra Combat Simulator in Chapter 17 but it is only unlocked after completing all Corneo Colosseum challenges in Chapter 9/14 and 13 Shinra Combat Simulator challenges in Chapter 17. On lower difficulties it’s unavailable. If you’re missing the Aerith solo challenge in the Corneo Colosseum then reload Chapter 9 and accept the challenge IMMEDIATELY after defeating Hell House. This is the only time in the game that Aerith is available at the Colosseum. If you proceed too far, she will leave the party.
The Pride and Joy fight takes place in the 5th round of the 14th and final challenge, ‘Three-Person Team vs. Top Secrets’. In a slightly bizarre twist, Pride and Joy is arguably the easiest of the 5 rounds, it’s the journey to that fight which is the hard part. You have to face Shiva, Fat Chocobo, Leviathan, Bahamut (who summons Ifrit to aid him at half health) and then Pride and Joy. The 4th fight in particular is easily the hardest fight in the entire game. Luckily, your HP and MP are healed 50% between every round.