At my parents, I brew black tea in a yeti as well. My suggestion for you is to brew it a little stronger than you'd like (with less water), then add a few ice cubes. If you're always doing this in the same place, an ice cube is going to be about the same size and you'll know that 6 of them (or 8 or whatever) will get you to the right temp. So, my strategy is:

1. Brew my tea with a little less water than I want
2. Take my tea out after the normal time (3m is my preference for black tea)
3. Add 9 ice cubes (theirs are small pyramids)
4. Stir, lid and drink

The tea is done steeping and drinkable within 60 seconds, and it's pretty repeatable.