Where are you in the Uk? We have managed holidays in the Netherlands on a small budget. Quit school 2 days early (we are lucky our school anyway plonks 3 inset days at the end of summer term). Use Tesco vouchers to get 2/3 off the price of a Le Shuttle crossing or the Stena ferry from Harwich to Hook of Holland. Couple of nights in a stopover location - we stayed near Ghent when we travelled via Calais and stayed a few nights nearby. Ghent tourist office has a free scavenger hunt for different buildings round town my dd aged 8 loved. Or if you go by ferry don’t miss Rotterdam - Plaswijkpark cannot be missed, it’s a full day out with all the fun value of a theme park without the queues and expense.
Then say 5 nights glamping at a Eurocamp. If your kids are good travellers, de Schaatberg looks good value (less than £700 and free access to pools and Mattel play area which is your activities sorted!).
Self cater and drive off the Eurocamp site to find a supermarket.
Do laundry at Eurocamping so you can minimise your luggage. The kids will probably love the launderette. Don’t forget to take string to make a washing line in your accommodation.
It would be tight for £2k but you might manage if there’s a bit of flex in your budget and you plan the heck out of it.
We are going back to the Netherlands this year with my dc 4 and 12, this time 12 nights, via Shuttle, staying in Bergues, Bruges and Eurocamping for 7 nights near Efteling theme park. We have budget this year so the travel accom and petrol alone will be £1800 but Bruges is a luxury. Eurcamping will be cheap - except a day out at Efteling theme park. For the stops on Bergues we will visit the town and local parks - you can often find a gem. I have found one that does a 1 hour pirate horseback stunt show for <10 Eur per head, for example and the res of the day will be a picnic and playing in the park.
if you plan something like this, research is your friend. And note city stops where you have to budget parking or public transport costs to visit city centres. Easily overlooked.
Also assuming your 3 yo is out of a buggy now I strongly recommend taking your kids scooters (teach the 3 yo if they can’t, and take a piece of rope to pull it along). I left clothes at home to fit these in last time and was very pleased I did!