Just some more information on the actual battle. Be sure you pack about 15 revives, 20 Hyper Potions/Max Potions, and 10 Full Heals.
The part where you fall short is probably right in the beginning, against Pikachu, seeing as half of your team is weak to Electric, and is probably KO'd by Thunderbolt in one hit. Open up with Mewtwo to ensure Pikachu goes down in one Psychic.
Against Venusaur, keep Mewtwo, since Psychic hits super-effective.
Against Lapras, use Floatzel's Bulk Up and Brick Break combo to take down Lapras easily (beware of Body Slam, which causes Paralysis). If Floatzel faints, then get Mewtwo into battle (yet again) to hit it with Thunder.
Against Snorlax, its Achilles Heel is its low Defense. Get Feraligatr in there to hit it with some powerful Waterfall shots. It should KO it in 3 or 4 hits. Waterfall also has a chance to flinch, so you may get lucky and completely disable Snorlax for a turn.
Against Blastoise, it has incredible bulk, so send out Floatzel and Bulk Up for a few turns (use Potions and such if his HP gets low). Then hit it with Crunch. The added Defense will defend against any Physical Attack that Blastoise might be packing, and Floatzel also resists Blizzard and his Water attacks. If Floatzel faints, then send out Feraligatr. If he too, faints, then send out Mewtwo for Thunder.
Charizard will go down easily to Floatzel, Feraligatr, or Mewtwo. Waterfall and Thunder will punish it.
If I could give you any last advice, it'd be to level up Honchkrow more, and get Floatzel up to about lv.60. Also, if you have it, get a Steelix to set up Stealth Rock in the beginning of the battle (It sets it up perfectly, and resists Quick Attack and Iron Tail from Pikachu, and is immuned to Thunderbolt and Volt Tackle). This will inflict damage to Red's Pokemon whenever they enter the battle (And against Charizard, who is doubly-weak to Rock, it knocks its HP down halfway, just by ENTERING battle! It also takes 1/4 of Lapras' HP)