Goodness, most parents would love it if their kids wanted to be out and about and not on games consoles/ internet/Netflix all day... yet he actually wants them to be LESS active. How bizarre.
Some good suggestions already made. Possibility of him going part-time, and you working a bit more to balance it out sounds very reasonable. He can enjoy house and garden more, which might ease his fear of not being around to enjoy it. Less work stress/slower pace of life might actually bet better mentally and physically for him, so he's less likely to obsess about dying too young AND less likely to actually die before he retires!
It does sound a bit like this fear of suffering the same fate as his dad has got out of hand.
£20 a week is not much to spend on two children in the holidays. Granted, they don't need to go to the cinema every week and there are lots of free activities around, but he has to realise that times have changed and things are different to when he was a child. I wish more children WERE just happily playing in their gardens or the street and parks but the way we live now has changed all that.
I don't think a shopping trip with lunch with DD is at all unreasonable. It's mum/daughter time and clearly not something you do often. As long as the shopping itself wasn't a huge splurge, I can't see the problem.
I think your DH has unintentionally allowed this fear of dying early to take over his life a bit, and can't see how it's impacting on you on the children. Doesn't sound like he's an intentionally mean or controlling man, rather he's developed a bit of a phobia that's influencing his behaviour. He needs to address this with you and together you can hopefully make some compromises that will make the situation better for the whole family.