Who emerges victorious?
That would depend.
Against an ordinary eagle with a bucket of water, I would say that Batman has the advantage.
If, however, the eagle and bucket of water are in fact the Wonder Twins, I would say that Batman has a MAJOR advantage…
Now, if it were an eagle with a jar of pickles…
That might be interesting.
“Wait a minute! How come I only get to transform into something lame like ice? How threatening is a bucket of ice water, anyway?! I could be defeated by a sponge! And it wouldn’t even have to be an evil sponge!”
God, I love that bit.
Put Zan up against Spongebob Squarepants. That’s a fight I’d pay to see.
Eagle loses. He needs a turtle instead of a bucket of water. Then he beats Batman.
Only if Batman is a bald, Greek philosopher.
So Batman goes up against the Wonder Twins.
But he’s not prepared…
I wouldn’t matter - all Batman would need is his batsponge and he could take out the other wonder twin with his bear hands. 
…I think “Bucky” could defeat the Wonder Twins. In his present state.
Bear hands?
What, he’s wearing a Wookie costume now?
The Wonder Twins vs. Wendy and Marvin…
Batman vs. the Wonder Twins? Oh, man, that’s not even fair. Batman would win with both hands duct-taped behind his back, and little pieces of duct tape over his eyes, and the Wonder Twins could have a spear.
FISH
In what country does this take place? If it’s the U.S., wouldn’t the eagle win since he’s got all those laws protecting him? Unless, perhaps, the eagle is a clumsy SOB and falls into the bucket of water, rendering him flightless until he dries.
Eagles are not rendered flightless by being a little wet. Birds have a gland that makes an oily substance that they groom into their feathers to keep them neat and relatively waterproof.
So should we correct that to: “Batman vs. an oiled eagle and a bucket of water”?
Now if it was a greased eagle, old Bats would be in deep trouble. Unless he brought his Bat-Birdly-Degreasifier 
If Batman threw one of his batarangs at the eagle, she might attempt to mate with it.