You sound very organised. My advice is to keep skiing throughout the summer, including the little ones, it gives them a huge advantage.
Because it is so late - like this year, you will need to pick a resort that is high up. I am presuming you mean Europe?
Places in the Alps like Courcheval, Tignes and Sainte Foy are good, check which way the slopes are facing, as that will make a difference to the snow, some in Sainte Foy are North and there is a glacier near Tignes, so doesn't matter if the snow is rubbish, just go there, it isn't the prettiest of place though.
Val Thorens is also a good resort for late snow.
Austria may be cheaper and we prefer the ski schools, but can be poor snow. Obergurgul and Zell am Zee are good, again there is a glacier nearby if all else fails.
That said, all the resorts can make snow, the only problem being by lunchtime it is often slushy and hard to ski on, although if you haven't been before, you may be ready for a rest then anyway. Probably a good idea to pick somewhere with a swimming pool, or little town, so there is something else to do.
Talk to everyone about what they prefer, there are so many options, DIY, hotel, chalets, sites like Snowheads are useful for reviews and to post questions.