If you look at many of the commercial power analysers (ours included) - they use DFT.
You will need something bigger than an 18F though.
Most commercial power analysers synchronise their sampling to a specific multiple of the fundamental frequency of the incoming mains. Either by using a PLL, so the digitisation occurs at the exact desired multiple of the mains frequency, or by digitising at a fixed frequency, assessing the precise fundamental frequency from the samples (e.g with a DSP based PLL), and then digitally resampling the signal at the exact desired multiple of the mains frequency. This means that when they perform the Fourier transform the mains harmonics sit exactly at the centre of specifc transform bins. If they arranged the sampling rate to be a multiple of the mains that works well for an FFT (e.g. an exact power of 2 multiple), they use an FFT. If they arrange the sampling to be at some other multiple they usually have to accept the higher compute load of a DFT.
Although it is easy to interpolate between transform bins, it is not easy to do that with great precision. Power analysers are typically trying to measure the harmonics to 0.1% or better. At this accuracy life is much easier if you align the bins. If you are only interested in the general makeup of the harmonics, rather than their precise magnitude, you can simply use an FFT at any old sample rate.