Nice work detecting pulsations at this RV level with the LHIRES and separating them out from the orbital component (I was slightly worried when I saw your RV and light curves in phase as The RV curve in Cepheids is normally approximately antiphase but then I noticed your light curve Y is plotted in increasing magnitude)
RV measurements can be tricky with the H alpha line because of its width. Narrow metal lines are easier, particularly if you can measure a few of them but then in the absence of telluric lines, a stable reference is difficult to arrange. I used two Si II lines with lamp lines superimposed on the spectrum during the exposure to follow pulsations in Deneb for example.
https://britastro.org/video/13862/14769
at 18:30
which suggested a long term sigma precision of 0.5km/s is possible
Cheers
Robin