Force Construction - The Batcave costs 85 points and is assigned to a character when you build your force. This character can be assigned the Utility Belt resource with a Utility Belt Costume and up to 6 Utility Belt Items at no cost. The Batcave can be assigned to additional characters for 10 points each. The Batcave adds Batcave Computer to your force. The Batcave can be played with a cost of 100 points; if you do, The Batcave adds Batcave Computer and Alfred Pennyworth to your force. The Batcave can't be assigned to either of these characters.

Since there are no keywords on the batcave/batcomuter, if I were to play a theme team with it, would I lose theme?

The Batcave Computer and Alfred are considered characters. Since neither have keywords, your force would not be a themed team.

This is a minor thing I wish WK would update, but mbstoner is right. Due to the computer and Alfred not having any keywords, it is impossible to make a theme team with the Batcave.