You will likely need more than $1500 to live well in Argentina, especially if you are thinking of putting your kids in an elite school. Also, your rental depends a great deal, as mentioned earlier, on whether or not you can rent temporary, or with a garantía. I recommend the garantía way if you are going to spend a great deal of time down in Argentina. Also, I recommend you consider Zone Norte, which is where I live. It's a lot more tranquil than the city, and you can actually get a decent sized house in a nice area with a backyard at $1000, and you have the river and other nice amenities close. I live in Olivos and in a car can get to Palermo in 15 to 20 minutes. It's outside of the capital proper, but in my opinion, much nicer. If you are working from home, you don't need to live in the Capital and go downtown on a continual basis, and a lot of expats just know the capital, and not anything outside. My quality of life has taken a quantum leap since I moved out to the north. You also have some of the best bilingual schools there.
Some of the actual numbers people mentioned here are pretty realistic. I imagine with $3500 to $4000 you could live okay out in a house to the north of the city, have a car, and if you're careful, have some savings from that, while sending the kids to a decent bilingual school (but if you send them to the best bilingual schools you will pay out the nose).
However, I recommend that right now, you wait for at least 6 months before trying something like that. The reason why is that socially things might get ugly down in Argentina in the short term. You don't want to be down here when they are burning stuff in the streets, rioting, and looting. Also, crime tends to increase quite a bit in these kinds of situations. I wouldn't recommend to anyone just this moment to even come visit with all the recent instability. It might turn out okay, but it could also end up similar to 2001. I might be going overboard here, but judging by my wife's reaction, who lived through 2001, I would be extremely careful. Don't put your kids in the middle of that, and my wife and I are ready to travel or relocate at a moments notice if things are super ugly.
Well, we would definitely do a garantia. We already maintain a 2 bedroom/2 bath apartment in my wife's hometown (12,500 pesos per month) that she secured with a garantia using her family's homes.