Childcare wise, if you want to discount the time he spends at work, which might seem fairest, he should be spending 64 hours per week in charge of the baby which does include night times.
So your 2 hours in the morning = 10
2 weekend nights + 16 (26)
Plus let's say one weekend lie in where you can sleep/relax/have time for you in morning until 12ish + 7 (33)
That's only just over half of 64.
So let's say in the evenings 4 days a week he takes over from about 7-11, to let you eat/clean up/relax - she's probably in bed for most of this anyway. + 16 (49)
One day in the week where you swap a full night's sleep for the early morning + 6 (55)
Then 9 hours spread over the rest of the weekend, of which there is 14 hours unaccounted for - the end of the morning lie in to the start of the bedtime shift. (64)
Just as an example of how he could take a fair share of childcare. By this I mean during the times he's "on duty" he does all nappies, clothes changes, all feeds/meals that come up, any routine stuff like bathing, takes charge of entertaining even if he does not entertain directly, does things like nail cutting if he notices, makes a note that new clothes need buying, washing, etc, and does the immediate cleaning-up-after which comes with taking care of a baby. ie, everything you do.
Housework wise, if you discount the time he's at work (40h) from the time you're both likely to be awake (112h) he's at home for about 64% of the time so half of the housework generated in that time is about 32% or slightly less than a third.
To simplify, that means if you're splitting housework fairly, for every two jobs you do, he should be doing one.
I would guess that roughly the kinds of things you're (both) doing every or most days would be as follows, and again I've roughly split them up, to give you an idea of what might be an example fair split.
Washing into machine (you)
Washing out of machine to be hung/dried (you)
Washing out of drier/off airer and put away (him)
Breakfast made (you)
Breakfast cleared away (you)
Washing up done x2 and put away OR dishwasher filled, dishwasher emptied (him)
Kitchen counters wiped down (you)
Bottles sterilised if relevant (you)
Lunch made (you)
Lunch cleared away (you)
Toys/baby things tidied up (him)
Dinner made (him)
Dinner cleared away (you)
Living room tidied, hoovered (you)
Bathroom wiped over (him)
Post dealt with and filed (him)
Recycling rinsed and sorted (you)
Then not every day but at least one of the following:
Change bed sheets (you)
Ironing done (you)
Proper clean of bathroom (him)
Take bins/recycling out (you)
Hoover bedroom (you)
Mop kitchen floor (him)
Garden maintenance (you)
House maintenance/DIY (you)
Car maintenance (him)
Food shopping (him)
Errands like car MOT, returning library books (you)
Clothes shopping for necessities (you)
Obviously different arrangements work for different families, but if you were to genuinely split things fairly, taking into account the time he's at work presumably working and of course contributing to the household, this is one example of how things can look. I hope that it gives some perspective to your question about whether you're asking too much and whether his current input into household stuff is really over what should be expected. Because I think that it's easy to lose sight when you're overwhelmed by it all or perhaps see other people's husbands doing even less. None of this is any more than 50% so it's not an unfair ask and yet (at least on the childcare side) it seems so much more than he is offering.