Since I posted this I learned about something called a Thiele Tube. It's a piece of glassware ($50-$100) that can be used to find the melting point of a substance.
You clamp this tube and put a heat source on the side opposite the hole (it's an oddly shaped piece), fill the thiele tube about three quarters full with oil, using a second clamp to clamp a thermometer holder and a small piece of rubber tubing cut into a ring to attach your sample capillary, submerge into oil, and slowly heat. The oil increases in temperature and heats the sample, it circulates around the tube to allow for slow and uniform heating.
There are more details you need to learn about if you are actually going to do this, but it seems very simple and easy. The major drawback I see is that the heat source in this case would have to be an open flame.