Nothing is quite so humbling as someone coming in and showing you that you are making your problem way more complicated than it really is. Lee pointed out that I was overthinking it: that a simple 2N2222 would work just fine, and indeed, I went back over it, and realized he was right. I entered in the schematic shown below into LTSpice….

And it looks like it will work fine. I had thought that the 2N2222 would have to dissipate too much power, but I made mistakes based upon two problems: I had placed the current limiting resistor on the emitter side of the circuit, which meant that the base of the transistor was improperly biased, and I had originally used a 12v supply. In the configuration above, less than 50mw gets dissipated by 2n2222, and just about 1/2w by each of the current limiting resistor and the LED. Cool.

And thanks Lee, for taking the time out of your day to lend advice to such an introductory topic.