You can easily cut down the eating out, that’s totally within your control. We had a bit of a takeaway/eating out habit until recently and now we do it once a month, saved a fortune! Much cheaper to eat at home.
I don’t really batch cook aside from pasta sauce which I make in vast quantities and freeze. To cope with the fussy child I’d just plan for indulging him - make the plain pasta and chicken, have a sauce with yours. They grow out of it. Mine is now 10, at age 5 existed on dust, air and plain pasta. She’s just started being much more adventurous with her food.
meal plan a repertoire of say 8 meals that the family will eat and do them on a rota, then you can buy the ingredients in bulk. For instance I always have plenty of tinned tomatoes, tomato purée, flour, lentils, beans, frozen onions, frozen veg, rice - you can knock up quite a few things from that.
for snacks I just buy a certain (small) amount and when it’s gone it’s gone. Tbh we aren’t big snackers and like to eat raw veg so it’s not a huge deal for us?
then there’s all the usual things - shop once a week, don’t go to the local shop for top ups etc. By being disciplined you can save money, a bit of planning really helps (especially with the eating out issue). I aim to do one very big shop at the beginning of the month to include cleaning stuff, then small top ups weekly of things like yoghurt, milk, fresh produce. If I get to the end of something before the end of the month I just do without, for instance I buy a certain amount of cheese per month and that’s it, if it gets eaten in the first week so be it, it’s not the end of the world!