While your live twitter idea seems a good one, the character limit appears to be cutting off a lot of updates and some simply don't make much sense due to this.
I had an idea that someone there may want to consider and code. How about taking each update from the Live team and running it through a script that replaces certain words with short hand versions that are more twitter friendly.
For example "Nico Rosberg" = "ROS", "Mark Webber" = "WEB", McLaren = "McL", Fastest Lap = "FL". That way more of your updates stand a better chance of fitting into the limit.
I'm not suggesting replacing the normal live updates in this way, just the twitter version of each update. The more characters you cut out, the more chance the twitter update will make more sense to the end user.