Lam, Lee and Shimozono recently introduced backstable double Grothendieck polynomials to represent $K$-theory classes of the infinite flag variety. They used them to define double $β$-Stanley symmetric functions, which expand into double stable Grothendieck functions with polynomial coefficients called double $β$-Edelman--Greene coefficients. Anderson proved these coefficients are $β$-Graham positive. For vexillary permutations, this is equivalent to a statement for skew flagged double $β$-Grothendieck functions. Working in this setting, we give a tableau formula for vexillary double $β$-Edelman--Greene coefficients that is manifestly $β$-Graham positive. Our formula demonstrates a finer notion of positivity than was previously known.