For this project, it sounds like you really don't need a lathe... You could cut the tapers yourself using a spokeshave or chair devil. A nice one will run you $65-100 new, but you can probably find one on eBay for a lot less.
Don't know how you're going to cut the spiral. The problem there is that you need to move your cutting tool at an absolutely constant speed. If you don't, then the spiral gets wonky. A metal-cutting lathe has this capability; it's how you make custom threads. On a woodworking lathe, I suspect it would be very difficult. Any chance you could get by with a series of evenly spaced grooves? If you need a spiral, I suspect it might be easier to draw it out by hand and then carve it out with a gouge than to try to do it while the lathe is running.