The flip side of this, though, is the OP seems to expect him to contribute 50% despite earning far less (except for luxuries she graciously treats him to) and hasn't mentioned how he'd be funded (as the lowest earner paying a larger proportion of his salary towards costs) in his period of shared parental leave.
It's really hard to see who is wrong and who is right here, but I can say clearly that if I was earning less than half of my partner and being expected to pay half of essential household costs, I'd be very, very reluctant to commit to taking on childcare that would mean a loss of my earnings, because it is clear that in that scenario that the higher earner saw their salary as theirs and not family money.
So all income in the pot, bills paid, joint savings topped up, joint (and child) spends accounted and paid for for, remainder split equally for equal spends each regardless of earnings. Once that is done, then it's reasonable to discuss who drops hours at work (with compensation for loss of pension) and what childcare is needed etc, safe in the knowledge that both parents will be financially secure.
On £38K a year, it's reasonably likely that it makes financial sense for OP to be the breadwinner and go to work full time, and him to be the care giver, maybe working part time to keep his skills up to date. And of course chores would fall mainly to the one who works outside of the home, since the SAHP is working 24/7 compared to the workers 8/5. But for sure, the SAHP needs equal access to money plus their pension topped up. And no-one on Mumsnet would be advising a woman who's higher earning partner expected her to contribute 50% to quit/pause their jobs to be a SAHM.