I think this is a good idea and you might find the hearty little happytime can run without a timing change, just the same way a common briggs lawn mower can. It might not be the most optimal setup, but probably will work fine.
The problem of oil injection could be solved by using the gasoline system to inject a mixture of oil. Someone asked if ashless two stroke oil exists. Yes, and its very common. It is sold commonly for marine 2 stroke engines that use oil-injection systems.
I would mix the oil with alcohol or window wash (methanol and water) so it could also help cool the cylinder by converting the water to steam. Surely, with a thin enough oil like the marine oil, you could just use the smallest jets available for HT carbs and just adjust the needle as necessary.
Since a happytime can get 80mpg or better, using a small jet or "lean" setting, it should run a long time on 1/2 gallon of oil mix and a tank of propane fuel.
I don't understand why one poster said it was "twilight zone" stuff or the thread should be closed. He should go buy a pre made bike and not worry about us tinkering types.
I don't see why anyone thinks the propane would "blow up" the crankcase. Gasoline does not blow up in there. The fuel is compressed and pushed through ports and then into the combustion chamber. It seems a small possibility that you could have a backfire in the carb.
a related thread here: http://motorbicycling.com/f37/propan...tml#post245006