What kind of image does Sir Gawain and the Green Knight try to convey through the beheading game?Since we know that both Fled Bricrenn and Sir...

Both stories mentioned have a great deal in common. There
is the "beheading game" in both, as well as a hero that honorably responds to the
challenge made by the giant/churl. (Fled Bricrenn or is found in
Celtic mythology, while the Gawain tale is set in the court of King
Arthur.)

I believe the image being conveyed in
Sir Gawain and the Green Knight is one that reflects the integrity
and chivalry of a true hero.

While the killing of the giant
seems an easy task to the untried Sir Gawain, when he does not win as he expects—because
the giant has the magic ability to pick up his missing head and walk out—instead of
fleeing in a panic, Gawain remains steadfast.

Sir Gawain
accepts that as a true knight he must answer the summons of the Green Knight as they had
agreed prior to the "beheading game." So one year later, he packs up and travels to
answer to his end of the bargain, a "swing for a
swing."

The image here is that of assuming the
responsibility for one's actions, or keeping one's word, because it is a question of
duty, not of one's personal preference. An average man might run away, as is the case in
the Celtic version of the story, but Gawain, not only a true knight, but a member of
Arthur's court, must fulfill his promise—it is the only honorable
choice.