Husband wants me to wake sleeping baby

[Valais]

We live in a building with a shared washing machine and only one weekly laundry slot, which is today (in two hours' time). Our baby is sleeping on me and DH is refusing to go and buy the laundry card from the concierge because it is my "turn". He wants me to pass the baby to him (and likely wake her) so that I can do it. I think he's being ridiculous and said no. AIBU?

Not to mention, if she just gave birth not too long ago, he should be picking up the slack anyway, doing a bit extra to clean up around the house, with meals, LAUNDRY, etc. ESPECIALLY if she's breastfeeding and/or pumping and therefore likely doing all of the night feeds/wake ups by herself. He better get used to doing things he doesn't like to do, but needs to, and get off the whole "bUt It iSn'T mY tUrN!" bull or he's going to be a **it parent and even worse partner. Trust me, my man, it will NEVER be equal because you'll never be able to "pay" back all her efforts for growing, birthing, and caring for this tiny human.