I could almost write your post OP. My DH does a 12 hour night shift with a 4 on - 4 off pattern. It's a killer. I work full time. We make it work. I think the key here though, is understanding. Sometimes it is an easy shift to make, from daytime sleeping to night time sleeping, other times it doesn't work too well and my DH just ends up sleeping and playing catch-up. You have to be considerate of that.
Fortunately for me, my DH finishes at 7am, so does the school run, however, by the time he gets home at 7:30, the kids are dressed and ready for school. He settles our DS for his morning nap and walks the dog while taking the kids to school. Then comes home, has half an hour wind down time, then goes to bed. He then sleeps till around the time I put the kids to bed.
I don't expect my DH to sort out the kids or do anything particularly (or potentially) stressful on his working days. This means that apart from the school run, I sort them out. I do expect him to walk the dog, and unload, load and turn on the dishwasher before he goes to bed. This means I can cook when I get home. However, this is the maximum I expect on his working days. It seems that popular opinion is that my DH perhaps does too much, however, I also work full time (teacher) and can not clean up after 3 other people on my own. To be fair though, he has gone to work today without touching the kitchen as he has struggled swapping this time - so I am going to have to do it.
Additionally, how much extra work is your DD bathtime? My 5 year old is really good at running her own bath, getting in, bathing herself and attempting to shampoo her hair (I always rinse it out but her hair is very fine). She then gets out and gets ready for bed. It only takes 2 minutes of my time. She does that while I sort out DS bedtime clothes or sort out clothes for the next day.
I also agree with the other posters that have suggested you lower your standards a little. Would be great if you could hoover every day, but would it really matter if you skipped the middle day of his 3 shifts? This would give you breathing room.