Can New Horizons be used to measure the distance to Betelgeuse (despite its fickle photocenter)?
In principle yes, in practice no. The telescope is good enough, but the CCD camera will saturate, preventing a good positional measurement.
Salient facts. The parallax to Betelgeuse as seen between Earth and New Horizons will be about 250 mas. The current distance uncertainty is ~20%, so to better that, the position of Betelegeuse measured by New Horizons would need to be more precise than ~25 mas.
Looking at the LORRI camera, it has a plate scale of 4.95 microradians/pixel and a point spread function with FWHM of 1.8 pixels, which equals 1.84 arcsec (Cheng et al. 2007).
A useful rule of thumb is that your positional accuracy in a resolved image is given approximately by
$$ \Delta \theta \sim \frac{ {\rm FWHM}}{{\rm SNR}}\ ,$$
where SNR is the signal-to-noise ratio of the image. Using this, it suggests that if one could get an image of Betelgeuse with a SNR of around 100, then you should be able to centroid the position to better than 25 mas using the New Horizons data. Combining this with a simultaneous Earth-based image (which one presumes would be much more precise), then the parallax could be determined to better than 10%.
The problem appears to be the brightness of Betelgeuse. Cheng et al. also give a photometric calibration formula for the number of analogue to digital units per second, $S$, recorded by the camera for a point source with a given $V$ magnitude.
$$V = -2.5 \log S + 18.94 + 0.4\ ,$$
where the extra $0.4$ is probably appropriate for a star of Betelgeuse's red colour.
Using this formulae we see that $S = 2.16\times 10^8$ ADU/s, assuming $V\sim 1.5$ for Betelgeuse.
The CCD camera on LORRI only returns 12 bits and has a frame transfer time of 13 ms, which means that 13 ms is about the smallest exposure time you could use and hope to get accurate positional information (Cheng et al. 2007), though in principle a 1 ms exposure time could be done. Even in 1 ms you would get $2.16\times 10^5$ ADU from Betelgeuse spread over perhaps 4 pixels. Given that a 12 bit ADC saturates at 4096 ADU, then Betelgeuse is more than 10 times too bright and will produce a completely saturated image whose centroid cannot be determined with any precision. If it were possible to remotely reset the gain on the CCD (currently 22 electrons/ADU) to something much larger, then maybe it would be possible.