Someone has mentioned the extra effort you will expend in chopping and cooking whilst breastfeeding.
Personally I think that is relevant.
I remember having to eat once an hour myself so that I could do breastfeeding. And I still lost weight...
Would go with the suggestions to change supermarkets:
There are at least two things which would bother me about your budget.
Since covid started I have tried to put some things by in case we have an emergency and run out of something. I'm not talking stockpiling, just looking at the prices and when there is a special offer buying a bit more and storing it.
Not only does this provide a sense of security in case you are ill, heaven forbid, but it is amazing the number of times we have run out of something and not needed to go to the shops as we have it in the store.
Generally now I aim to have two of everything in, aside from fresh fruit and veg of course.
Aside from that, I would be concerned at DH setting the budget and it not being a joint budget. If he never does the shopping he is likely to be out of touch.
Try using Quidco for cashback on everything you buy or similar.
Through Quidco there is a company called Approved Foods which features products where the box is damaged and so on but products are within date.
Fourty pounds delivery only 1.99 delivery charge I think.
Baby stuff on there.
Basically it is a company which takes boxes etc of stuff that bigger supermarkets get ridof
I've personally had some really nice things on there.
Also try mixing veggie mince or generally veg with bolonaise etc.
Sticking to the budget you have got is a lot of work I find to set it up.
and you are already 'working' looking after two kids, bless you.
Perhaps when you want a holiday you could ask your DH to do the shopping within that budget for two weeks. I expect he wouldn't be able to manage it.