When set up properly, I managed to get a cheap SIS derailleur to shift perfectly well, even on an old freewheel without the shaping common on modern sprockets to aid shifting, though this was a 5-speed with wider spacing. 'Better' derailleurs just seem to be lighter, and more expensive.

With $100, you could probably pick up an almost-roadworthy bike at a yard sale and get it on the road again as a 'beater' for use in winter or in dodgy areas if you don't already have one. Depending on what parts you have already, I'd spend money on: brakes if the current ones aren't up to much, decent fenders if you don't have them already, a different handlebars if you don't like the current ones, lights, cycle computer etc. If you have all of these and everything works, save the cash or spend it on something else you need.