Which star with a Bayer designation is the farthest from the Sun? The key to finding the answer seems to involve processing data from astronomical databases, which I'm not sure how to do in this case. I'd prefer to limit the answer to the original list of approximately 1500 stars, but answers that rely on a list that contains stars with more recently acquired designations could be interesting to consider as well.
See important caveat to this answer below

At a distance of 2992 light years, Tau Canis Major is the furthest of the 1513 stars that have a Bayer designation, with several others almost equally distant. I computed the full list of Bayer-Sun distances: https://github.com/barrycarter/bcapps/blob/master/ASTRO/bc-bayer-distance.csv

Important Caveat

I used the HYG catalog at https://github.com/astronexus/HYG-Database but it may be inaccurate. In particular, Wikipedia gives Tau Canis Major's distance as 5,000 light years, although I couldn't find this number in the source provided (https://doi.org/10.1088/0067-0049/199/1/8), although I only skimmed.

The far from canonical https://www.universeguide.com/star/taucanismajoris notes that 2992.32 is a revised figure, down from 3197.68, but, of course, the figure may have been revised again.

Methodology

After downloading the HYG database file, this one liner creates bc-bayer-distance.csv:

zcat hygdata_v3.csv.gz | perl -F, -anle 'if ($F[9] &lt; 100000 &amp;&amp; length($F[27])&gt;0) {$F[9] *= 3.26156; print "$F[9], $F[27] $F[29]"}' | sort -nr | tee bc-bayer-distance.csv

A brief explanation:


The 9th field (starting our count with 0 as Perl does) is the distance, but there's a note on the HYG git page: "A value >= 10000000 indicates missing or dubious (e.g., negative) parallax data in Hipparcos". It turns out the actual number is 100,000 (I've submitted a correction), so we ignore distances greater than 100,000.
The distance is in parsecs, so I multiply by 3.26156 to convert to light years.
The 27th field is the stars Bayer letter designation without the constellation. If it's empty, we ignore the star.
The 29th field is the stars constellation name, which, combined with 27th field, yields the Bayer designation.