Grocery shopping is genuinely terrifying at the moment isn't it?
My advice would be start by purchasing the things that are non negotiable to last until the end of the month. Its 26 days if you literally mean the last day of the month-baby formula lasts 5 days at £14/each- so 6 of those to be absolutely sure, £84 off your budget straight away-£266 left. I think this is completely doable as a one off for the month.
You mentioned the dog as well, so factor in dog food costs and then work out what you have left after essentials like bread, milk etc.
Loads and loads of nutritous, tasty and filling evening meals can easily be made for £1 a head or less-bulk out bolognaise or chilli with pulses, grated carrot, rice.
Batch cook corned beef hash/soups.
Choose cheaper cuts of meat/downgrade labels if you can.
Don't be tempted to try things like too good to go if you havent used them before-they are good but I find you need to be experienced in adapting meal plans/supplementing with ingredients to make these worth while & also in a position for it not to matter if they are a bit rubbish (it happens!).
When I have been on the tightest of budgets, I have literally meal planned every meal and done a dummy online shop to make sure I am on budget.
It absolutely can be done with meticulous planning, you've got this!
Edited because my spelling was atrocious