Naice
In school holidays
£2k budget
Pick 2.
If you could go as late as possible in August, it would cost less and the weather would still be hot.
You say you don't want to self cater, but that's your best option here if you don't want to compromise on time of year or accommodation quality. Then you have control over the food. Unless you pay ££££s for HB/AI it will be a buffet and also possibly not to your taste.
SC will probably be cheaper to book separately, but also worth looking at packages, especially from jet2holidays, who have a lot of free child places, if the airports suit. Don't get hung up on 4 star, it's fairly meaningless and mostly relates to facilities not 'naiceness'.
Be flexible about where you go, it sounds like pretty much anywhere in Portugal, Spain (incl islands but bear in mind that eating out in the Balearics is a little more expensive IME) or Greece will suit. But pick somewhere that has a dishwasher and is walking distance to restaurants, beach and supermarket then stock up with breakfasts, drinks, snacks and easy food and eat out at least once a day, then the 'catering' element is minimal.