You might want to try the Tea Map (
http://www.teamap.com).
Unique So Chique, at the corner of Wilson and Magnolia, does a good high tea. You wouldn't confuse it with the big hotel tearooms (the interior is cozy, not grand, there is no live music, and you won't feel like you've stepped into a Wodehouse or Delafield novel, but all the funny characters are elsewhere, leaving you are stuck with the aristocratic bullies and the maiden aunts ... who disapprove of you in vague and unsettling ways).
UsC offers a variety of teas --- black, herbal, green, and white; the service is good; and the tea has the traditional mix of sweet and savory, with good scones, sandwiches, and pastries. Prices are much lower than the hotels but, as mentioned above, the experience is different. The tea is served in a room in the back of a small shop, so it meets your private room requirement.
You do have to get past the goofy name, and get up to Uptown, but I think UsC would do a good party for you.