Yeah, I've been there, too. Fortunately, they won't be in the brooder forever and if you can get them outsider sooner rather than later, life will get better.
I don't think you need enough water for the ducklings to play in while they are in the brooder. I just made sure their waterer was big enough for them to get their beaks in the water fully. Still, they will splash water all over the place.
I remember cleaning out the duck brooder a lot. I used wood chips in the brooder for my chickens, and did not have to clean it out the whole 8 weeks they were in there. But ducks are not chickens and ducks just require more maintenance in the brooder.
A couple options to think about... I think I would use wood chips for ducks, and have something under the waterer to catch most of the water they splash out. Something like maybe a pie tin, filled with wood chips, so the top of the pie tin is level with the other wood chips in the brooder. When they spill water, maybe it would be mostly confined inside the pie tin, which would be easier to take out and clean than the entire brooder.
Another option I am think of is using paper shreds, which work great for my chickens, but I don't imagine that paper shreds would last very long with ducks. The good thing for me, is that I can shred my own paper at home for free, so when the paper shreds bedding gets wet and needs to be changed, I would just gather it up and throw it into the compost bin. Then dump some fresh paper shreds in the brooder for them. I don't know how long the paper shreds would last with ducks, but at least they are free and if you don't mind changing the bedding maybe a couple times a day, then you might save some money.
I hope you are exaggerating that their bedding is only good for about 10 minutes. That would make it a real challenge. I don't know where you live, but I can tell you the second time I got baby ducks I made sure it was warm enough to brood them outside! Learned my life lesson with ducks from the first batch.
BTW, I recommend you update your icon profile with your geographic area. That helps members give you better advice if they have an idea of your weather. My setup in northern Minnesota is likely a lot different than someone in southern California, for example.