Per Kovetz Bais Aharon V'Yisroel in an article written by Rabbi Aizik Ausband he writes that according to those that Mitzvos do not need Kavana, when you are singing Tzur MiShelo you are in essence Bentching - therefore it is best not to say it. He also mentions that The Gra also wrote not to say it and it is not printed in his Siddur.
In the next paragraph it seems that Tzur MiShelo was sung at Shalosh Seudos (the 3rd meal) and since at times people are not really hungry then, Birchas HaMazon is only MiDrabanan.
Regarding the 3rd meal on Shabbat (Seudat Sheleshit or Shalosh
Sheudos), since there are opinions that one doesn’t need to eat bread
at that meal – one would not repeat the bentching if he forgot the
addition. And, in regard to Yom Tov, this is certainly true – since
everyone holds one doesn’t need to eat a third bread meal on Yom Tov.