If you don't want to mess around with cages, keep in mind that you can throw the sponge around with a minecart. Example of a blueprint for doing this:
Z-1:
WWWWWWWW
H..D..XW
W.WWWWWW
Z-2:
WWWWWWWW
RRRRRXHW
WWWWWW.W
Z-level of the sponge is Z-0. W is wall, D is a door, . is floor, R is an impulse ramp, H is a hatch over a channel downwards linked to a lever elsewhere. Yes, both hatches need to be linked to levers.
The sponge is above the tiles labeled X, which are walls in which you'll have to dig an upwards ramp: first on Z-1, by a dispensable miner with the door locked behind him, then on Z-2.
Once the sponge has fallen down to level Z-2, put a heavy minecart onto the hatch on the Z-1 and pull the associated lever.
The minecart falls onto the impulse ramps, which accelerate it; five ramps(but better make it eight or ten, water makes impulse ramps less effective) should be sufficient to make the sponge fly back and onto the second hatch.
The second hatch leads to the place where the sponge is supposed to go, which, I assume, is a multiple-Z-level area with a pool at the bottom.
Benefit of this method: doesn't require you to mess with pumps and terrariums after the sponge is caught. Hatches react immediately, so you can micromanage the timing to drop the sponge in without letting much water through. The rest of the construction is going to be drowned, though, so make sure your levers are elsewhere.
That said, I'm only about ninety percent sure it'll work, as I tested the concept in arena mode and didn't know how to spawn impulse ramps there, so back up your save just in case it fails.