The freezing point and melting point occur at the same temperature. You simply call this temperature the "freezing point" if you are going from a liquid to a solid (by lowering the temperature of the system) as you pass it, and you call this temperature the "melting point" if you are going from solid to a liquid (by raising the temperature of the system) as you pass it. So the point where the two curves intersect should be the answer, and it is.
My guess is that the experiment was designed this way on purpose because the plateau for lauric acid may not be as easy to discern (given your experimental setup) as it was in your previous experiments (as you say the change in phase might be too quick to notice clearly based on your experimental setup). Nevertheless, you do see a gradual flattening of the freezing point curve starting at about 43 right? Even if you did not have the second curve to help narrow things down you can still sort of see that the point of interest lies roughly near 43 from the first curve data alone. The second heating curve just helps you locate things more precisely.