Do you have a bike coop or shop with a good selection of used parts nearby? At $2-400, you do not have the luxury of picking and choosing parts, but with some hunting you might find ones that work really well. Befriending the right person at one or more of these stores could get you a long ways.

Edit: old 5-speed rear wheels will work. You can shuffle the spacers from right to left and to get the freewheel to line up with the crankset, then "re-dish" the wheel to center the rim in the frame. With a little luck, you will be able to do that at zero expense besides labor. This does require wheel truing skills. That same coop could help here.

Ben