I didn't have any luck with that so I thought it was a bogus strat, I'll give it another go/.
My lineup and gear and strat:
Tifa: Mythril Claws, 4 slot bangle, fury ring +magic and MP spec, cure+magnify just in case, ATB assist, lightning, fire
Aerith: Silver Staff, 4 slot bangle, fury ring, +MP cost reduction + magic +MP, ATB Boost, First Strike, Lightning + Elemental, Fire
Barret: EKG Cannon, 4 slot bangle, Gotten-whatever it's called that gives me limit break from the start, +MP +HP, +Luck, First Strike, Lightning+elemental, Time materia
Shiva:
Lay down Astral Ward with Aerith, build up 2 more bars, and have Tifa build 2 bars. Whoever's closer to the Astral Ward, get in and start spamming firagas. Shoot Focus Shot with Barret when you see pressure state. If you have high enough magic stat, you'll kill her in 6 or 7 fireballs without even having to use the limit.
Fat Chocobo:
Build 2 bars with Tifa, get level 3 chi. Build up 2 bars with everyone else too. Use overpower + regular attack to put Fat Chocobo into pressure state and use focus strike and focus shot. When you stagger Fat Chocobo, Tifa should be at 2 bars of ATB ideally.
Right before you stagger Chocobo, start Aerith's Judgement Ray.
Stagger Fat Chocobo do Tifa combo True Strike, True Strike, Rise and Fall, Omnistrike, attack to 2 bars ATB, True Strike, True Strike to put the stagger damage multiplier over 300%. Somewhere in the middle of that combo, or even before you start it, activate Catastrophe with Barret.
Leviathan:
Build 2 bars with Barret and Aerith with an Astral Ward. It's tricky to get opportunities to use Thunder, so you need to use Tifa as bait to force Leviathan to do its close range attacks. When it does shoot off a Thundaga in the Astral Ward with Barret and Aerith. Repeat this cycle one or two more times until Leviathan is pressured. Once he is, start nuking with Thundaga, and Focus Strike to stagger. Right when he staggers, immediately use catastrophe since Leviathan's stagger time is really short. If you have ATB to spare, get in some True Strikes with Tifa to increase stagger damage multiplier. Hopefully you kill him before he starts phase 2, but if not, keep nuking with Thundaga and shooting with the ranged characters.
Bahamut:
Use Focus Shots and Focus Strikes to try to get Bahamut's stagger bar up as much as possible before he summons Ifrit. When he's almost at 50% HP, get Aerith in the Astral Ward and do the firaga, ATB boost to get max ATB, firaga, firaga combo. Get Tifa in there too and have her do Firaga, Firaga. Hopefully you did it in the right order so that Aerith gets a little more ATB from Tifa's ATB Assist materia. This should put Bahamut into a fallen down pressure state, and if he's in this state, he won't summon Ifrit. You don't have a timer to know how long this pressure state lasts, so use your best judgement. Use Barret's Catastrophe too. Hopefully it kills Bahamut before he summons Ifrit, but it hasn't happened for me yet.
Pride and Joy:
This thing is a Pushover, just make an Astral Ward, and spam Thundaga all day long with all 3 characters.