Two hours is quite restrictive, that narrows it to Spain, Portugal, Balearics and Tenerife etc.
4-6 hours would get you further to better weather. i know Egypt is exceptionally cheap this year following the riots in Cairo last year. But the red sea resorts are miles away and you can get some great 4 or 5 star all inclusive deals for well under £1500, same in Tunisia, although it is a very resort based holiday.
We are hiring a villa in Tuscany for £1500 peak season with it's own pool and child friendly. So for Sept it'd be around £600-900 for a private villa in Italy with pretty ok weather?
A couple of weeks ago we stayed in the Altovincino Family village in Lake Garda, which was like an Italian Haven. We paid 200 euro for 4 nights in 2 bed chalet. It had a kids cub, creche and baby pool, water rides etc. Was ok actually. They have other sites around Italy near cheapy Ryanair airports.
Or there are similar campsites in the south of France? Set up ready to sleep luxury tents and caravans/chalets?
We're planning on taking our DD to Sardinia next year (she'll be 8 months by then) September weather will be beautiful there, it's a fantastically child friendly island, not too busy, the island is beautiful and you can get cheap flights from pretty much anywhere in the UK. Last time I was there the accommodation was as cheap as you like as well and there's a good mix of country villas and town apartments available if you want to self-cater. I don't know anyone who hasn't fallen in love with the place once they've been.
we went to Italy when DD was 6 months - it was lovely, not too hot and plenty to do when the little ones need to get out of the sun - recommend Sorrento as so many places to visit from there: Capri, Ischia, Rome, Positano, Amalfi etc - we loved it so much we got married there the year after. We went 5 years ago and paid £1200 for 2 weeks half board in June so i would imagine similar prices in sept.