Definitely agree with the PP who said a small resort won't be enough once you can ski. We've been twice. Both times to the same tiny (but beautiful) resort. We're going to La Plagne next as we feel the need for something bigger.
What are you looking for and when? If you're not fussy about accommodation and are open to anywhere, then I managed to price up for a family of 4 at Easter in France at £912 on Sunweb. This includes equipment hire, lift passes, and accommodation in a 1 bed apartment but not lessons, food, or transport. I'd say about £500 in fuel/crossing costs and another £1000 in spending money.
If you can go outside of the school holidays, that will be cheaper. It'll be cheaper to drive than fly (though don't forget you'll need to buy chains or socks). Easter is a bit cheaper if you are restricted to holidays.
Your big expenses will be lift passes and lessons. France is more expensive than Austria on both counts but more accessible to potentially drive, so it could offset some costs.
I've not been to Bulgaria, but I hear it is cheap for a reason, and I've not heard great reports.