As the stage notes indicate, their contest, a contest over exacting the revelation of the other's secret as well as the other's affirmation of the repressed, is symbolized by their struggle over Brick's phallic crutch. The crutch is a "weapon for which they were fighting for possession." No longer Brick's support, it appears as the means by which the men would do injury to each other.
Brick and Daddy's final struggle thus marks the reverse side of the narcissistic love between them, the aggressive logic of "either you go or I go" between those who mirror each other too closely. Note in this respect Daddy's desperate response of Brick's revelation. Brick will certainly outlive him and he will have to pay for his coffin.