That's conservative. A123 batteries I've used in my model helicopters are said to be able to go from full to empty at 35C and can manage 60C for short periods. There are Lithium Polymer batteries on the market that are claimed to be good for 150C or more. In practice they probably won't manage many cycles if you do that to them with any regularity, but it wouldn't surprise me if you could get 60C reliably and A123 cells are pretty bombproof.
What would you actually need to be able to do? Climbing to 300 ft then accelerating to 140mph needn't take too long. You'd need enough reserves to maneuver for a bit then land pretty much immediately if required. It sounds feasible to me. If you put a 250hp motor with 3 minutes of endurance in a sub 200kg 1 person aircraft, it should do 0-140 and 0-300 feet pretty quickly.
You can pedal a fuselage like this up to 80 - 90 mph on a kilowatt or so (human power + gravity with a 2/3% gradient) so presumably you could get it up to 160mph on 10 horsepower or so - without wings.
How big would you need the wings to be to attain a stalling speed of 140mph? How much induced and parasitic drag would they have? This is the part of the question that I really don't know how to answer, but I imagine if you were to design an aircraft to fly at high speed without any requirement to be able to fly at low speed (because this is taken care of by the VTOL system) you could make them considerably more efficient. I suspect you might have a better handle on this than I do, Genghis?
Perhaps given a more modest cruising speed you might be able to power such an aircraft very efficiently - perhaps just a few horsepower to fly straight and level at 100+mph? Obviously a practical aircraft needs some excess power to outclimb downdrafts and outclimb terrain. Also, most people find being in an HPV claustrophobic. However, I bet you could find a compromise hybrid or wholly electric vehicle that would give you a vehicle with VTOL capability and a half-decent range, and that it could end up looking not altogether unlike the Lillium.