HM, I'm no expert............
I would say that 3 days is part-time and 4 days is not. Four days is doing without you for a day and squeezing 5 days into 4.
Your husband's earnings have no bearing on your working life, and it is nobody's business to comment on his earnings and that effect on your family and, frankly, it is vulgar to bring it in to conversation. 'We' are not discussing his earnings, 'we' are discussing your working hours and finding a happy medium between the life/work balance to the satisfaction of you and your employer.
Your decision to work is not (I'm guessing) financially motivated but personally motivated and it is that that is the tricky bit. I fail to see the importance of where your nursery is in the world.
I think they have to be assured that when you turn up to work to do your 3 days you will be fully committed to that time. Legally I have no idea how you stand with a verbal agreement.......(I know in Scottish law a verbal agreement over, say, buying a property, is binding).
I would be inclined to sit back and think through how you want to approach them. Then ask for a new meeting to discuss it further and politely and calmly state what you would like and why and why they will benefit.
Before then though I guess you have to face the big question of......if they will not back down, do you love (and need) that particular job enough to do 4 days.
If the answer is yes then do not feel bad and grab it. If the answer is no then be prepared to throw in the towel.
Look at life from a different perspective. Weigh up how satisfied your working life makes you and how satisfied your children do. (Think ahead too, your children are constantly changing.)
There is no right and wrong here.