Following up on what I said about how Post-Vulgate Gawain should utterly despise Arthur - having absolutely no reason to like him - there is an additional context from the Post-Vulgate itself that I forgot to mention in the last post and should be considered:
Gawain will kill you even if you're his kin
(Also, Agravain, as his usual self, will gleefully join in the crimes if he can)
So yes, Blood Relation will not matter to a murderous Gawain, especially if the slight was done when he was a young impressionable kid (King Lot's death was when Gawain was eleven, if you recall). By Post-Vulgate's own logic, Gawain should itching for an opportunity to kill Arthur himself.
Coincidently, this cements another reason why T. H. White's The Once and Future King's Villainization of Morgause (and Morgause being evil in general) is a very BAD idea: it removes the last vestiges of any reasonable justification as to why Gawain is even a Knight of the Round Table - or even a hero! - in the first place.
In the Medieval Narratives, the reason why Gawain and his brothers are part of Arthur's Court in the first place is because of Morgause' insistence, in defiance of her husband. Morgause is a supporter of her brother, not another one of his enemies.
Without Morgause putting in a good word for her brother and actually scheming to destroy him, Gawain and his brothers will be right there working with their mother all the way.
Ironic, as one of the reasons why many modern writers love to villainize Morgause is so that they can reframe Mordred's incestuous birth as some sort of evil scheme to usurp the throne (and in the process, absolve Arthur of the responsibility). In reality, that's a really, really impractical, nonsensical and completely redundant rationale - Gawain, Agravaine, Gaheris and Gareth ARE Arthur's closest male relatives and therefore, legally in the best position to take the crown. And all without the taint of incest, an abominable act that would disqualify Mordred from inheritence if made known.
