What will it finally take to accurately measure the distance to Betelgeuse?
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.