We explore KLIP forward modeling spectral extraction on Gemini Planet Imager coronagraphic data of HR 8799, using PyKLIP, and show algorithm stability with varying KLIP parameters. We report new and re-reduced spectrophotometry of HR 8799 c, d, and e in the H and K bands. We discuss a strategy for choosing optimal KLIP PSF subtraction parameters by injecting simulated sources and recovering them over a range of parameters. The K1/K2 spectra for HR 8799 c and d are similar to previously published results from the same data set. We also present a K-band spectrum of HR 8799 e for the first time and show that our H-band spectra agree well with previously published spectra from the VLT/SPHERE instrument. We show that HR 8799 c and d show significant differences in their H and K spectra, but do not find any conclusive differences between d and e, nor between c and e, likely due to large error bars in the recovered spectrum of e. Compared to M-, L-, and T-type field brown dwarfs, all three planets are most consistent with mid- and late-L spectral types. All objects are consistent with low gravity, but a lack of standard spectra for low gravity limit the ability to fit the best spectral type. We discuss how dedicated modeling efforts can better fit HR 8799 planets' near-IR flux, as well as how differences between the properties of these planets can be further explored.