I think what you're aiming for is feasible, but batteries are the main issue. There is only one type of LiFePO4 cell around that will give you the very high maximum discharge rating you need, A123. The snags are that the cells are small, so you need lots of them, plus A123 won't cell individual cells in bulk. To get them you need to buy power tool packs and dismantle them, unless, like the Killacycle team you can persuade A123 to sponsor you. (see here for details of the Killacycle:
http://www.killacycle.com/)
Rather strangely, I'd recommend looking at ordinary lithium polymer cells, despite the safety issues (which are primarily about charging). Yours is a specialist application where weight and maximum discharge current dominate, capacity, life and absolute safety in a "normal" vehicle environment are less important. RC modellers have been using LiPo cells at very high discharge rates for years and have learned how to deal with the safety issues.
LiPo will also give you a higher system voltage for a given number of cells, as LiPo cells work at about 3.7V rather than the 3.2V for LiFePO4. The cells will also be a lot lighter, as they will be pouch type flat packs. Kokam seems to be a well-respected brand; they make cells that will discharge at up to about 30C, which is on a par with the A123 LiFePO4 cells. Most other LiFePO4 cells seem limited to about 5C to 10C maximum at the moment, hence my thoughts that LiPo might be better for this application.
To deal with any safety issues, it might be an idea to consider a battery pack that could be fairly easily jettisoned, or at least has a quick release for rapid removal. Although a battery fire is very unlikely, a lithium fire is an exceedingly unpleasant event by all accounts. An acquaintance who has built an electric paramotor is using LiPo cells with no problems, but has taken this precaution of building in a jettison system.
Motor choice comes down to looking for those with the highest torque and speed range for a given weight. I would hazard a guess that a pair of the big Lemco 200 or Perm 132 PMDC motors might be an option. Both can run at a few hundred amps for short periods, both weigh around 10 - 11kg each (pretty light for their power) and both would give similar peak power outputs (around 15kW each, perhaps a bit more if over-run). Although it doesn't sound much, 30kW of electric motor power in something this light will be pretty quick.
System voltage will need to be up around 96V or more, to give you the speed range you need.
I don't think you need a gearbox or diff/transaxle, just hook up one motor per drive shaft via a couple of big chain drives. You could consider fitting some form of slipper clutch connecting the two drive shafts to allow better traction off the line, by allowing both motors to drive one wheel if traction is poorer on one side. I'm sure the torque and speed range of these motors should be OK for the hill climbing. In my old hill climb Clan Crusader I never used anything other than first and second gear, although I frequently ended up with the rev counter banging on the 12,000rpm stop.............
Good luck, it sounds like an interesting project.
Jeremy
I've got a Ginetta G27 by the way 
which I built too