Discharge rate of the batteries you can't really know until you know their capacity (assume they are half their original capacity, and just grin at any extra you get), *and* the actual power usage of your system in your actual conditions (which we can guesstimate, but it requires numbers from you to do this).
If you use small batteries at a very low rate, they discharge slowly and can last a long time. If you use them at a high rate, they discharge quickly and last much less time.
I doubt that for your broadly-stated range goals and conditions that a $250 battery would get you anywhere near what you want.
To help you figure out how much battery (or whatever) you'd need, you would need to tell us the actual range you need, at the actual speed you need that range at.
Wherever there is anything other than perfectly flat roads, you also need to know the slopes, so you can figure out how much power it really takes to go somewhere at a certain speed. Note also that rough roads take more power, as do sandy conditions or other conditions where tires sink in instead of staying on the surface. Anything where the tire doesn't just push you along the surface, but instead has to move some of the surface out of the way, will take more power. How much more...depends on the surface and the vehicle/rider weight and the speed and the tire type/size/etc.
If there are winds, then they add to your speed when riding into them, and subtract when riding with them, and crosswinds will probably add to your speed in one way or another. A 5mph headwind means that riding at 10mph is actually taking the same power as riding at 15mph. A 20mph headwind means riding at 20mph is actually like riding at 40--it will take so much more power you probably won't be able to maintain your speed.
So there are quite a few things that affect power usage, and that all affects range. It also affects whether a given system can even do the job you may ask of it--it might take a lot more power than that little motor system, or the little gas engines, can deliver.
With numbers, a generic wh/mile figure can be guesstimated, which then lets you figure out how much battery it will take to go a certain number of miles at those speeds under those conditions.
Without numbers for speed or desired range or slopes/etc, I could only guess that for "miles and miles and many miles" and "as much range as possible" you would need a huge trailer full of battery or a trailer full of generator to keep the small one constantly charging (or directly powering the controller via the generator). I'd guess it will
This doesn't include going up hills or mountains; I doubt the motor could handle much of that, with the trailer full of battery or generator.
I don't know how well those little "chinagirl" style gas engines can do that either; or how reliable they'll be in the dusty and dirty conditions you'll have there, especially if it's not on smooth roads (they seem bad enough here in the city in Phoenix from my limited observations).
I guess the good news is that it's still a bicycle so even if the motor fails or runs out of battery (or gas) you can still pedal it.
If you want to see how the range vs speed vs conditions / etc thing works, you can play with the simulator at http://ebikes.ca/tools/simulator.html , and the trip simulator linked from there.
The same thing applies to the gas engine; it's just not simulated there, and you'd need to use a conversion page to see how much gas is equivalent to a wh of battery.
You can guesstimate that a 1hp gas engine will be about equal to a 750w electric motor system (but they aren't necessarily equal because it depends on how each is driving the wheel and the way it's being used, and under what conditions).
