I assume BASS estimates the resolution in a similar way to ISIS by estimating the FWHM of the neon calibration lines, in which case it may not give a good estimate of the resolution on undersampled spectra like this. (Not enough points in the line profile)
There is still something odd about the dispersion value. The native dispersion of the LHIRES 2400 at H alpha is ~12.8A/mm so for example giving 0.083A/pixel with my ATIK314 (6.45um pixels) Your native dispersion should be around 0.06A/pixel so it looks like there is a further 3x binning going on somewhere. In any case I suspect you are losing some resolution compared with other LHIRES III 2400 spectra of the same object. Compare for example the cleanly resolved metal/telluric lines in similar spectra on the ARAS forum as here for example.
It seems that H alpha is likely to be well covered so I might take a look in the UV. With the potential for variation in size of both the H alpha disc and the M star. It could be important to establish the extent of the M star independent of the disc. It is a more difficult area to work in but I had some success in this area with the similar but much fainter eclipse of AZ Cas
There is a discussion involving Phil Bennett about this area here