I ebf our twins four years ago, and am currently ebf our 8 week old.
To be honest, when the babies are little, I have done everything for them on a day to day (and night to night!) basis, though DH would do the odd nappy change/burp/cuddle and took them out for an hour in the pram on Sunday mornings so I could soak in the bath. But the other 99.9% of the time, babies were/are my responsibility. This was a natural situation to me - he wasn't here between 6am and 8pm Monday to Friday, and I was the only one who could feed them.
DH picked up a lot more of the 'other' responsibilities in the home though. I did/do what I could during the day, which often wasn't very much when at home with the twins, and DH picked up the rest when he got home/at the weekend. At the time, I thought he resented the extra work and thought I was slacking during the week, but in hindsight, I was projecting my own concerns and guilt. I thought I shod be AE to run a ship-shape house, cook from scratch, feed and raise new twins, and do lots of maternity leave activities! In hindsight I can see that that was a bit ambitious!
During the last eight weeks, again baby is my responsibility. Partly because I'm breastfeeding again, but also because I am at home day and night and get used to being responsible, while DH is back at work 6am to 8pm again. I can't stop planning/thinking about/listening for baby, and equally DH can't just turn on that focus for two hours when he gets home! So again, I am doing baby, and DH has to pick up what I can't do. He is very very hands on with the older two (the twins he had little responsibility for during their first year!) - getting up for them in the night, sorting their breakfast at the weekend, taking them out for a few hours at the weekend so baby and I can sleep (ha ha! I wish baby would sleep on demand!).
I have rambled. But I think my message is that I basically do our babies almost unaided. But that doesn't mean DH does nothing, because he earns money, cooks, cleans, shops and entertains older children.
From your post on here, two things really stand out to me:
Firstly, he works from home, which I can imagine drives you mad when he won't just hold this or fetch that. But he is working - if he was out working in an office he couldn't help. Personally I think you need to stop asking him to help with baby in the day (though he could maybe stck the dishwasher while the kettle boils for his tea?!).
Secondly, there is the tennis four nights a week. I read on here once that it is futile to try and work out who does more/less domestic chores/work by looking at the chores - much easier to look at the amount of free time each person gets. So I would tackle your husband from that perspective - you need equal time 'off' even though you're unlikely to go out during yours because you need to be on hand to bf. If he has a night out doing tennis, maybe you'd like a night in without responsibility for baby (have a bath, read a book, sleep, paint your toes, pop out for a run ...). He is being very unreasonable if he won't facilitate that on an equal basis, though I guess he is absolving himself of it by offering to pay a babysitter.