Are some people saying because her dh has an early start when he gets in he should do nothing? And can't be expected to help look after his own child?
My dp does shift work, long hours including early starts, evenings and weekends, nights etc. But when he is at home he is as much 'on duty' as I am.
If one of us is knackered the other will do more, but he would never expect to come in from work and do nothing whilst I do everything just because he has been at work.
We have no idea what time either the op or her dh go to bed, maybe he stays up till 10pm? Not every adult needs 8 hours sleep a night and as the op says a the child is having sleep issues maybe she isn't getting that either?
From 3:30when he goes to work she so still on duty, she may get to sleep or she may be dealoign with the toddler sometimes.
They both need to work out when they can have some time off, of they want a rota the can have one, but its not unreasonable for him to help get their daughter in bed etc even if he has been at work.
What is the new plan op, is he wanting to go to the gym every evening? That would be unreasonable, but you could both have a night pof the week when you can get to exercise, even if its just a half hour run etc. Then Sundays are split if thats how you want to do it.
We dotn do fixed days, times like that, we can't anyway because of DPs chamgign shifts but also I wouldnt like the rigidness of that, some days I may be fine but some days I might really need a break and the same for dp.
You need to find a balance that works, but there is no reason why working means you get to opt out of parenting once you are home.