You need more varied bedding as described by
@aart. You may benefit from raising everything up a bit too. For example, make the bamboo sides taller, fill in with chips/bark, etc to a higher level. Then rain drains through and down further and the sides keep the chips in better. Next, you should shield more of the sides, focusing on the sides that get any prevailing winds that tend to blow rain in.
we had to do this similar approach in our run. So.much.rain this past year, that we were left with poopy mud (yuck and it stank!) so we got a huge load of chipped wood and raised it by 8-10 inches (once it settled). Had to put boards around the fence sides to keep it all in. Then we added to the uncovered side of the run: coarse sand, peat moss, sawdust to make a nice and loose soil for dust bathing.
cute set-up though!