Is this man a child or an adult. Why are you making excuses for him. He is perfectly capable of getting up in the night, doing housework and childcare when he is home. He just doesn't WANT to.
What he earns is irrelevent unless it's enough to pay for full time 24/7 childcare, a cleaner, personal shopper, gardener, chauffeur, laundry person, etc? This is the value that you currently add to the household so please do not let him try the old 'my work is more important than yours' whine.
I don't have to be in an office suited and booted every day would be his argument - whereas he does - This is NOT an argument. It doesn't even make sense! Does he get annual holiday? Days off if he's sick? Weekends? What do you get?
What does he do for the 3 hours he is relaxing on public transport.
You are both busy from the moment he leaves for his work to the moment he returns. All other times - morning, evening and night - you should share the housework/childcare equally. He is not 'helping you out' he is just dealing with his responsibilities as a husband and father.
Stop taking his crap, treat him as the adult he is. For example, Ive stopped putting his clothes away but that just means they lie around for ever which drives me nuts - then don't wash them. Leave them dirty until he realises he has no clothes. He will then wash them, dry them, iron them and put them away himself. If he doesn't put them away, put them on his side of the bed. If they get crumpled he will have to re-iron. It won't take long before he gets the message.
Sheesh!