What will it finally take to accurately measure the distance to Betelgeuse? Comments below What is the maximum distance measurable with parallax? discuss challenges associated with parallax measurements of Betelgeuse and link to Wikipedia’s Betelgeuse; Distance measurements which contains the intriguing paragraph:

In 2008, using the Very Large Array (VLA), produced a radio solution of 5.07±1.10 mas, equaling a distance of 197±45 pc or 643±146 ly.83 As the researcher, Harper, points out: &quot;The revised Hipparcos parallax leads to a larger distance (152±20 pc) than the original; however, the astrometric solution still requires a significant cosmic noise of 2.4 mas. Given these results it is clear that the Hipparcos data still contain systematic errors of unknown origin.&quot; Although the radio data also have systematic errors, the Harper solution combines the datasets in the hope of mitigating such errors.83 An updated result from further observations with ALMA and e-Merlin gives a parallax of 4.51±0.8 mas and a distance of 222 (+34/−48) pc.[10] Further observations have resulted in a slightly revised parallax of 4.51±0.80.10
83Harper et al. (2008) A New VLA-Hoppocaros Distance to Betelgeuse and its Implications
10Harper et al. (2017)A Updated 2017 Astrometric Solution for Betelgeuse

Looking at section 3.1 of Harper et al. (2008):

As discussed in Section 1, Betelgeuse required a significant additional error source (cosmic noise) to obtained the expected χ2. The actual nature of this cosmic noise is not known but it may be related to photocenter movement which is unlikely to be random in position on short timescales and would likely provide a systematic error in the astrometric solution. The position angle of the star's rotation axis has been measured from spatially resolved ultraviolet Hubble Space Telescope spectra. Uitenbroek et al. (1998) found ~55° (measured East of North) from absorption features in Goddard High Resolution Spectrograph data, and Harper &amp; Brown (2006) found ~65° from emission features in multi-epoch Space Telescope Imaging Spectrograph data. The distribution of photocenters, perhaps driven by convective and Coriolis terms, may have a special relation to the rotation axis. The stellar proper motion vector also has a position angle of 68° and brightness fluctuations that occurred preferentially near the stellar equator might induce scatter in the parallax displacements, although fluctuations due to stellar rotation itself, would be unlikely because of the ~17 yr rotational period (Uitenbroek et al. 1998).

If I understand correctly; the problem with measuring Betelgeuse's position accurately is that it is so large and non-uniform that its &quot;photocenter&quot; (radio, optical, or otherwise) can be significantly different than its center of mass.
And the problem with obtaining accurate parallax determinations is that its non-uniformity changes on timescales comparable to a year. If it were much faster or much slower, it would not be so much of a challenge.
Interferometric optical imaging of Betelgeuse confirm dramatic non-uniformity across its disk.
Question: What will it finally take to accurately measure the distance to Betelgeuse?

To see how nonuniform Betelgeuse appears optically: (feel free to edit and add better links)

this answer to What else could the Event Horizon Telescope Observe?
this answer to How far have individual stars been seen by radio telescopes?
Why can&#39;t the surfaces of stars be observed? links to Wikipedia's List of stars with resolved images
This answer to Betelgeuse in false colors, is this common?
The problem is that the apparent diameter of Betelguese is about 50 mas (milli arc seconds --- 1 mas  is about 5 nano-radians) while its parallax is about 5 mas and its shape and surface brightness are both irregular and variable.
Given that, the current measurement are amazingly accurate. So I can identify about three approaches to doing this measurement, and I don't know enough to know which one(s) might really work, or which might be expected to work first.

Measure the distance without using parallax. This most likely means getting an accurate estimate of the absolute brightness or all, or part of, the disk of the star and comparing that with the apparent brightness. Spectroscopy might give an accurate idea of the temperature, which should in turn allow the brightness to be calculated and interferometry might allow the measurement to be restricted to a small area  near the centre of the disk which was not obviously a star-spot or anything.

Simultaneously image the disk and measure its position relative to background stars, allowing the position of the centre of mass to be estimated accurately.

Increase the baseline of the parallax measurement. A pair of telescopes on opposite sides of Saturn's orbit, for instance would have 10 times the baseline, and more or less ten times the accuracy.


Update: It seems the Gaia consortium now expects to work around the brightness constraints and get data for all stars. From the wikipedia page

Although it was originally planned to limit Gaia's observations to stars fainter than magnitude 5.7, tests carried out during the commissioning phase indicated that Gaia could autonomously identify stars as bright as magnitude 3. When Gaia entered regular scientific operations in July 2014, it was configured to routinely process stars in the magnitude range 3 – 20.[57] Beyond that limit, special procedures are used to download raw scanning data for the remaining 230 stars brighter than magnitude 3; methods to reduce and analyse these data are being developed; and it is expected that there will be &quot;complete sky coverage at the bright end&quot; with standard errors of &quot;a few dozen µas&quot;.[58]

I don't know how much that will help with Betelgeuse though, due to its large apparent diameter.