Honestly, I would deep litter it, and as swampy as it is now, you need stuff that will raise the level FAST. Any chance you can get a load of wood chips (NOT shavings, actually chunks of wood) or mulch in there? Then you can put the "nicer" bedding on top of that, but the wood will give it some bulk right off and raise them out of the muck, plus it will break down, but slowly, and pure mulch or wood chips doesn't really do anything gross when waterlogged, so it shouldn't get stinky or anything like that.
I would suggest 1/2 inch hardware cloth, or even landscaping fabric to hold the litter in without holding the water in.
It might also help to put some of those black plastic drainspout pipes (the ones with the holes in them every few inches) under the wood chips, so once the water works it's way to the lower levels of the litter it will go into the pipes and run out to the sides of the run and dump out on the grass (and the water will have some chicken poo in it, so your grass will grow like crazy after it rains)