I've added the Ice Warrior race to the original post. The ability is pretty straightforward - just note that it only applies to the game immediately after the loss, meaning it applies to a maximum of two games per match. It's a little weaker than the Cyberman ability, but I think Water makes up for this.
Below is a short list of possible changes I've ultimately decided against, and why.
Random support cards when a Time Lord regenerates - In the show, regeneration is a very random, uncontrolled process, so it would make sense to randomly swap out some of the support cards upon regeneration. While this is a nice connection to the show, it could potentially cripple a player if the RNG is unfavourable. If the randomly selected support cards are the staples of three decks, then those three decks are ruined.
Fewer support cards for Sontarans - I want every class to have the same number of support cards, to keep things a little simpler. I'm not too worried that Sontarans are overpowered, since they still need to rely on Gravity cards, even if they have arguably the strongest ability.
More upgrades for Cybermen - A very rough estimate of the duration of the event would be 9 rounds. By the last few rounds, the Cyberman will have become quite powerful.