I am wondering if a high pressure mercury vapor lamp can be more energy efficient and have a higher CRI if I use a triphosphor coating commonly used with modern fluorescent lamps as the fluorescent coating on inner portion of the outer bulb.

For example, if I use a /841 phosphor on a 400w high pressure mercury vapor lamp, will the lamp have a CRI of about 80 and higher luminous efficacy compared to ordinary yttrium vanadate phosphored high pressure mercury vapor lamps?

It would practically light just as a bare MV without any phosphor.
One reason is, the triphosphors are sensitive mainly to the short wave UV, but the MV emits most of the energy in the longer wavelength range.

And even with theoretically most efficient phosphor, the efficacy wont be better.
The main difference why the MV are so inefficient compare to the fluorescents is not the phosphor efficiency, but because of the arc efficiency alone.
The low pressure gives most of the energy in the form of the 253nm UV, which is what excites the phosphor.
MV gives most of the power as the long wave IR thermal radiation of the hot arctube.
Then the power is split between the blue, green and UVA peaks. Even if you convert all of the UVA into the visible (that is, what the phosphors used on MV already do; they aren't any less efficient than the fluorescent triphosphors, there just isn't enough UV power available to excite them), you barely get enough light to balance the color to CRI of about 60 by radiating in the orange range (still not that low eye sensitivity so reasonable lm/W figure, but the saturated red are still severely missing).
Higher CRI then means you have to suppress the green or blue peaks and radiate in the longer wavelength red, which means the efficacy would drop way further.

Unfortunately this does not work.  The phosphors used in fluorescent lamps are optimised for stimulation by the 253.7nm UV-C radiation, which is almost not present from high pressure lamps that instead radiate chiefly at 365nm.  Different phosphors are therefore used in the HPMV lamps.  Moreover the fluorescent lamp phosphors quickly lose their performance if they become too hot - again a different temperature class of phosphors is required for HPMV lamps, which makes things rather complext to achieve good efficacy and CRI.