OP. Congratulations on your lovely new baby.
I feel so sorry for you, my DH used up his limited paternity leave looking after our other DC whilst I remained for a while in hospital and very shortly after I was out had to return to work and I remember really panicking. What that must be like with a C section, I can't imagine but you have my full sympathy.
Practicalities.
You need another pair of willing hands.
If your family is not available... what about his family?
Hiring short term help may seem like an awful expense at the moment, But it is really important that you get help in these early weeks. Seeing someone else having to come in and be paid do things he could easily be doing might shame him into it.
Do you still have midwife visits? Or a health visitor. They are a fount of information and may be able to help you out with recommendations of where to find a mother's help or someone to come and babysit your toddler make them lunch eetc. even a morning's help would be useful. Tell them (privately if necessary) and ask for their advice. They've seen it all before and may know what works.
Also get the health visitor or midwife to TELL your DH specifically what you cannot do medically that he needs to help you with. He's not listening to you but at least he will hear it loud and clear from a professional. Again, they might shame him into it.
Its too tiring for you to have an argument at the moment but you could write down the toddler's routine and present tasks one at a time that you want help with. So that it sinks in. "Tomorrow, please could you do toddler's breakfast at 7.00 am. or Please Can you take Toddler to the park after their lunch?
His behaviour, the nonsense about training ( isn't that what you are doing now? getting the toddler used to having a sibling? or does he think you need to send them on a course? ) ALL OF THIS IS DESIGNED TO STOP YOU ASKING. It's designed to distract you with an argument to change the subject, or just give up asking, so he can go back to thinking about himself.
But keep asking calmly, reasonably for what you want. Even if you don't get it. Don't let his behaviour silence you. If he's not going to do it, at least he has to say out loud it's because he doesn't want to do it, rather than making you justify yourself. "I don't want to do it." "I appreciate that but I would really like you to." "You should have trained the toddler." "That may be, but we are talking about toddler's breakfast. Please do it for toddler."
Going forward, you said you have a well paid career. You will have more options if you can to hold onto it, and stay in the workplace,even if you are part time, wfh etc.