Condensed milk doesn't mix well into iced coffee
It definitely works if everything is room temperature or a bit warmer, maybe 20-30C (68-86F), and it definitely won't work if it's all pretty cold.

It sounds like you may be a little too far on the cold end. You have coffee at 0C, plenty of ice to hold it at 0C, and a small amount of condensed milk at room temperature that'll rapidly cool down as soon as it's added. The upshot is that you're basically stirring at 0C, not even room temperature, and condensed milk at 0C is indeed pretty darn thick.

To keep things simple, I'd start out by just stirring the condensed milk in without ice. You could even stir it into just the milk/half-and-half, since with a smaller quantity of refrigerated liquid, the temperature won't drop as much. Stirring that into the rest of it should then be much easier. You may also want to stir continuously as you're adding the milk, rather than after adding it all.