Honestly, I don't think that you can add that many more matches.
Let's just ignore the Dome's requirements and consider this on just the basis of running fields. Each field requires six radio channels, and runs on a six-minute match cycle. (In other words, we have six robot-matches completed every six minutes.)
The IFI
radio modem has forty channels. Six of them can be accessed by teams, which leaves the possibility that someone will make a dumb mistake and run off the field on a radio and lock up a competing robot. Clearly, this prospect is unacceptable. Therefore, we have 34 channels available to us. That breaks down to five fields of 3v3, or eight if FIRST reverted to 2v2. (We'll ignore 2v2; FIRST's current-generation field barrier is made for 3v3, no matter how Mission Mayhem rigged it up, and should be used by past experience through the 2009 season.) So yes, FIRST could run a fifth FRC field in Atlanta. Now how do you incorporate that into the elimination bracket on Einstein in a manner that is both fair and easily understood by the average person on the street? I don't think you can.
Allison made an interesting idea with giving each division the GTR treatment, but I believe a similar radio problem would happen here. You'd have to run both fields on the same channels, for the same reasons as above, which would mean that if any robot or OI was turned on on the inactive field, they'd be transmitting on the same channel as the competing robot. I haven't tried it when it's just an OI or RC, but I have empirical evidence to prove what happens when you have more than one robot connected and attempting to communicate on the same channel.
Since we can't really add more fields to the Championship, we've got to get in more robot-matches through some other means.
We could decrease the match cycle time, either by a shorter match or a shortened inter-match period. Shorter matches aren't as interesting--two minutes flies by as it is. Shorter intermatch periods mean both the teams and the field crew have to be absolutely on top of things. That six minutes involves running the match, having the refs score it, field reset doing its thing, any field issues being repaired, teams moving their robots off, the new robots coming on, getting set up, and ready (including a once-over by the IFI rep). If anything goes wrong, matches get delayed and go later than planned. Especially on Saturday, this is unacceptable as well.
This leaves the final option of lengthening the Championship. This involves either starting earlier or finishing later, both of which cost more, both in venue fees and personnel costs. (Don't ignore the cost of volunteers--food, T-shirts, credentials, and the oh-so-spiffy volunteer pins all cost money. Less than paying them, but it still costs money.) The other problem is that you get burnout here--volunteers get more exhausted, and some have long drives ahead of them that evening. Getting a crew to stay even longer when they have enough trouble staffing at the current level will require some solution I'm not equipped to produce. (I'm marketing, not management.)
It's a complex problem, one that would be interesting for any management science folks out there. But I really can't see a solution to it.