Not sure what I'm missing here.
Why doesn't he just look for another job? You either have enough money to survive or you don't.
Work out your incomings, your outgoings, what you have left and how long you can live off whatever is left over.
I suggest he starts looking for a job now. I am due my first and I would love my other half to be at home with me, however that's not realistic. £6000 would be gone very quickly.
You say you only really have gas and electric outgoings... what about council tax, car tax, insurance, fuel, food, phone bills, luxury items/social time, what about the outgoings you'll have once baby is born?
Eating out all the time because you are lazy and can never think of what to cook is just bizarre. I'm a bit surprised any adult simply 'can't think of what to cook'...
Sounds like you just don't want him to go back to work straight away and you want people to justify that decision. Also sounds like he needs to go back to work as let's face it, £6k isn't going to get you far is it...
Sorry to sound harsh but he needs to start shoving a load of job applications to agencies or local places etc and start looking for a job, grafting and supporting his family. I couldn't be with a bloke who knew I was going on maternity leave and was willing to drain our savings so he didn't have to go back to work straight away. Maybe a week or two off, but not when you are so concerned about finances.
Also:
I am  at anyone going out for dinner several times a week because they can't think of what to cook, having £6k in savings, but thinking it's appropriate to apply for benefits.
^ this - it does take the piss a little!