October or April AI with young kids is perfect.
In those months, if you want to spend your days on the beach/pool rather than wearing clothes sightseeing, I would aim for Canaries (any), Turkey (Antalya area is great for airport transfers; watch out for transfer times in other resorts), or south south south Mediterranean (Crete, Rhodes, Cyprus).
In my experience, the Balearics, mainland Spain, mainland Greece, mainland Italy and the northern Mediterranean islands are too cold for beaches/pool. There will be good years where it's warm in April/October, but not reliably so.
AI with kids is the best, particularly if you can let go of all the boundaries you have at home. I let my kids have whatever they want on AI (I think the record is 9 ice creams in one day!) and don't worry about it - not thinking about cooking or what anyone is eating is what makes it a holiday for me. I did let the 4 year old have fizzy drinks from the drinks machine, but not caffeinated ones!