... Since I need to buy a new MTB for all year round training, I would use it for the trip. I am interested in the 29er concept (2000$ to 2500$).
The TA has been done many times on mountain bikes and a few times on department-store bikes. One guy rode XC on a unicycle!
But why not a touring bike? Your budget will get you a good one, which is excellent for training and much more versatile than a 29er. It is fine off pavement (although not for cleaning logs and jumping off rocks) and a decent club ride machine when unloaded.
Consider that you will spend 4,000 to 4,500 hours on this bike doing the TA. Training an hour a day, five days a week for a year is only 260 hours. I'd spend my money on the main item.
You can learn a lot by browsing here, but get a good grounding and a summary from Schubert's article in the ACA archives at http://www.adventurecycling.org/resources/201004_TouringBikeBuyers%27Guide_Schubert.pdf
. The April 2011 issue is not yet in the archives, but if you can find a copy, Stephen Lord's "Find Your Way to the Perfect Touring Bicycle" covers the same ground from a different slant, and arrives at nearly the same advice.
The touring bike has reached its present form for very good reasons, which you will know after reading these. Of course you can ignore them, the main penalty being discomfort.
I expect the replies you get here will 1) try to get you onto a touring bike, or 2) try to get you to modify a MTB to become more like a touring bike. Do take advantage of all that experience and get the right tool for the job.