There are two separate issues.
He sounds fed up with the knee rocking thing so I think as a couple you should make a plan going forwards to wean her off that and/or find an alternative which works better for him. Or he could make the plan and ask you to support him with it - either would be totally fair.
Whether it's a fair share... I'd try this thought experiment and see what comes out of it. Preferably together - I think if you do it and simply present him with the results, he may feel attacked, which would be fair.
There are 168 hours in a week.
Subtract the amount of hours he's usually at work per day, plus his commute times to and from work, x5.
Subtract the amount of hours your DD reliably sleeps through the night, x7.
If she doesn't reliably sleep through, count her usual night time sleeping hours x7, and halve that. (Because someone has to be alert/on duty, but can mainly sleep/relax during this time).
Ignore things like lunch breaks, naps, leisure or social activities where one of you is out of the house.
The number you're left with is the amount of hours during which both parents are available for childcare. Round it to the nearest even number, it just needs to be a rough guide. Say you end up with a figure of 40 - that means there are in theory 20 hours each that could be allocated during the week if you were going to split things totally 50/50 - so as your experiment start to divide this up into bedtimes, dinnertimes, chunks of weekend, morning supervision, "night duty", etc.
Then - housework - work out how much you're each home, potentially making mess. Take your 168 hours per week again. Take off 8x7 sleeping hours, so 112 for you. Then for him subtract that work/commute time again.
Your figure is 112 and his will be lower. Say it's about 62 - roughly work that out as a ratio. So for 112:62, you're home close enough to twice as much as he is.
List the jobs which need doing in the house on a weekly basis to keep it running. This doesn't include childcare or meals for yourself/child only but does include family meals for everyone, grocery shopping, meal planning, laundry etc as well as more obvious things like filling the dishwasher, changing sheets, cleaning toilets. If something needs doing more than once a week just add a x7 or x3 or whatever. Just list/count the tasks up, no need to estimate time or effort, it's just a very rough reckoner here. Now your experiment is to split this list according to your ratio, so 2:1 by my earlier calculation which means you "take" 2 jobs off the list for every 1 he "takes".
Now you've done your experiment of what 50/50 childcare and housework looks like - compare it with what each of you actually tend to do.