If you are hellbent on doing it with piston, you can do it with fuel injection motors (I believe OS had fuel injected competition grade engines for many years now. They are priced as if they were made of gold though). Or as mentioned above, servo operated mixture (and possible automated, if you are into programming microcontrollers).
IDK how Hill did it, but said methods should make it possible. Hill was a skilled chap it seems, so I wouldn't be surprised if he had automated mixture control of some kind.
That would be a minimum requirement in my opinion to have any reliability at altitude with curburettor motors.

Its still not worth in in my opinion though, because of price and complication.
If it turns out that electric is not feasible, then go for turboprop.
Turbomachinery is in general a lot more efficient at altitude, and at extreme altitude its also the only thing that will run at all. Because it doesn't have a fixed compression ratio, it will compensate for the thin air by spooling, hence increasing its compression as needed to keep up to rated power.

Heres a price example aswell:

Fuel injected piston:
http://www.jhbol.com/store/index.php...cPath=64_65_70

Turboprop:
http://www.1classifieds.co.uk/-1/pos...ound-1100.html

The price is about the same, dollar for power. The turbo is twice the price, but its twice the power aswell. Plus, it has a lot of features included that the F.I. piston doesn't.

Turbo also has the added benefit of wetstart and integrated starter motor, so IF you shall have a flame-out (motor stops, in jet language) you can just restart it with a push on your radio, IN FLIGHT.
If the piston sucker stops at altitude (which it is a lot more likely to do) you will have to land the plane and restart it manually, on the ground.


That said, I still believe electric is simplest and best, but if it turns out you can't get enough battery into the plane to make electric feasible, then please consider turbo seriously before you go with piston.
Turbo has a lot to show for, for this purpose.