If he does nothing, then OP probably is having to be on call constantly.
And when you are doing all the childcare, all the nappies, all the baths and bedtimes, every early morning wake up, every night feed, washing every dish, putting them away, making every meal, buying everything, taking out every piece of rubbish, washing every item of clothing, drying it, ironing it and putting it away, doing all the dog walking, making and attending all the appointments, doing all the household admin, buying every birthday card and present, and the myriad of other jobs which often fall to the 'wife' - it becomes wearing. Like any job.
But when you are doing all those things, and there is an equal partner who refuses to lift a finger during his time away from work, then it becomes an untenable situation and yes, one I would suggest a separation for. Not a split, you notice. But if a conversation won't get him to buck up his ideas, then a separation might make him realise just how much OP contributes. And as well, it's not just a separation because he won't 'wash a few dishes', but it would be all about the fact that OP has appealed to him to make their lives more balanced, and he has either ignored her or doesn't care. That's not an equal, loving partnership.
(OP, not saying he doesn't care, hopefully he will step up to the plate)
I'm fed up to the back teeth of people (sadly often women!) minimising the contribution of the sahp. It is much more than bunging clothes in a washing machine and doing an online shop once a week.
Don't forget, if OP left him, he would have to do/pay someone to do all of these things, and her workload will be more than cut in half. Far better he steps up to the plate now.
What are your suggestions for OP's DP's days off wildbill, as they seem to have gone unmentioned in your post? Should the demands of the household be split 50/50?