If this is something you want to do once, I can't see it being worth the effort.
Working within a niche, there is no cheap plug and play setup!
However, I love that you are looking to provide a unique service.
If you can find a way to really package this, I could see events hiring you to shoot live during a show!
The images could also be uploaded live to social media, so that people can see how amazing the event is, and then decide to attend, increasing the turnout of the event.
If the photos are good enough, it could be a huge asset to an event coordinator.
What you are looking to do could be a real service!
To answer your question, I can only speak of the equipment I know that could make this a reality.
Newer Macbook (post spring 2011, I would go for an Air for portability), airplay-ready streaming out to an Apple TV.
I would carry an Airport Express (or whatever alternative) incase the WIFI signal is weak where you are.
Use a 6D or an Eye-Fi to wirelessly transmit the images you take.
This would be a 2 person operation, where all you do is shoot, and all they do is manage the stream.
Hahaha, however, if you are only looking to do this for a birthday as a one-off, then yeah.
I love entrepreneurial efforts, so I may have gotten a little ahead of myself.