Believe it or not, but each of your pokemon are very effective against Tate & Liza, You need Shuppet to burn Claydol to get the ******* to stop spamming Earthquake (because burns cripple physical moves), after that use Grumpig (because shuppet probably fainted) and spam psywave, which isn't resisted by psychic types and does a fair amount of damage in-game.
For Xatu I recommend having Swampert at the ready, who's tanked every previous attack so far (and spammed surf). I'm pretty sure this will keep Swampert labeled as the greatest threat and Xatu will target it until it's too weak. This gives time for Grumpig to go MVP, take out Claydol and/or Xatu, likely both if you keep using your items. If swampert fails then send out mightyena, this will toss out an Intimidate debuff and render it so that Xatu must attack Grumpig, who resists psychic moves.
Lunatone and Solrock, it's likely that you'll fight these 2 together. Lunatone will set up and up and up with Calm mind until it can sweep your team with psychic, but this leaves it prone to Psywave, dark types (mightyena), and burns/intimidate (I'm pretty sure it has Rock slide or Earthquake). Solrock is a chump, It's more physically attacking than Lunatone but I recommend burning it with will-o-wisp after Lunatone (because lunatone is an Donkey imo :L) and Vibrava resists everything Solrock can throw at it, and hit hard with crunch.
This is mostly a battle of budgets because you're dependent on revives with pokemon that low in level, and may be using lots of Soda Pop from Slateport beach or Hyper Potions from the Pokemart, I also am not sure if this is the best way to deal with Tate & Liza so I didn't mark it as an answer. Thanks.