Can the Batman and Catwoman move, then outwith, then use the duo attack?

Nope.

Assuming you are asking about the Double Pounce SP, you can not use Outwit in the middle of an action.

No. But after the attack they can outwit something else. Then next turn you can Outwit, move and double attack because they have willpower!

Yeah double OOTS! (not exactly but similar)

Say Bats/Cats has Maneuver, Lunge, and Vault, on the first maneuver, it can move the opponent up to three squares right? But on the second lunge/maneuver, only 2?

Vault

Prerequisites: Combat Reflexes or Leap/Climb

Choose a character.

After the character resolves a close combat action, it may be placed in any square in which it may legally be placed adjacent to a target of the action at the same elevation as the target.

Maneuver

Prerequisites: {boot}, Close Combat Expert or Combat Reflexes

Choose a character.

When the character make a close combat attack against a target opposing character, after the attack is resolved, put the target in any unoccupied square adjacent to the character.

Lunge

Prerequisites: Close Combat Expert or Leap/Climb

Choose a Character
When the character is given an action, but before it makes a close combat attack, it can break away automatically and move up to 2 squares.

Let me illustrate, t is the opponent, B C is Bats/Cats

x x x x x x x x x x x x x
t x x x x x x x x x x B C
x x x x x x x x x x x x x

use the Double Pounce SP to leap climb 8 squares up
x x x x x x x x x x x x x
t x x B C x x x x x x x x
x x x x x x x x x x x x x

lunge 2 squares to become adjacent to an opponent and use the first of the Duo Attack
x x x x x x x x x x x x
t B C x x x x x x x x x
x x x x x x x x x x x x

maneuver the opponent 3 squares over to the other side of Bats/Cats base
x x x x x x x x x x x x
x B C t x x x x x x x x
x x x x x x x x x x x x

lunge two squares over and use the 2nd of the Duo Attack
x x x B x x x x x x x x
x x x t C x x x x x x x
x x x x x x x x x x x x

the above is my question. That lunge can only be done that way right? I can't put the Bats side all the way to the right of C?

Maneuver the opponent two squares over
x x x B x x x x x x x x
x x x x C t x x x x x x
x x x x x x x x x x x x

Vault to the other side.
x x x x x x x x x x x x
x x x x x t C B x x x x
x x x x x x x x x x x x

But this Vault is legal right?

Vault dioes not work because it is not a close ACTION. But the rest is just fine. Manuever happens before the second Lunge. Manuever is after the attack is made, lunge is immediately before the attack

wow, I totally missed that Vault is an action and cannot be used with Duo Attack.... glad I asked.

now the question is I can Maneuver the target up to 3 squares over right? ... I'm pretty sure that's ok.

but when I lunge after that, I can't swing the other end of Bats/Cats to the other side right? That violates the rules for moving double-based figures.

your are correct on all accounts.

Yup, totally legal.
For even more fun, add in Sidestep for more movement.

... forgot about sidestep...thanks!

so remove vault and add in sidestep to get:

Bats/Cats with Lunge, Maneuver, and Sidestep.

So how will that work with sidestep? Bats/Cats moves up 8 squares to be 3 squares away, makes a ranged attack, move 2 squares using sidestep, makes a close combat attack, and maneuver? Then I don't need lunge at all. Or actually, I may not need sidestep.

If you want the maximum amount of movement then it would be:
move 8 squares with Leap/Climb
move another 2 with Lunge
make first close combat attack, including Maneuver
break away and move 4 squares with Sidestep
move another 2 squares with Lunge
make second close combat attack, including Maneuver
all in all... 16 squares of movement.

I thought you were a genius because of the Al avatar, but I know you're a genius because of this post. That's beautiful!

a break away roll is still needed because Leap/Climb is no longer in effect right?

if the break away roll fails, you can still lunge to the back and attack from there can't you?

think of it as a full move charge

Yes.
Nope. If you fail to break away, the action ends then and there.

I think I've got a good one. Batman & Catwoman have Combat Reflexes, which allows the figure to take knockback from any attack, but as a double-base figure, it cannot be knocked back.

So, based on the existence of the errata on Mister Miracle & Oberon's TK SP, I would guess that this needs errata for them to accept the knockback.