  We introduce families of two-parameter multivariate polynomials indexed by
pairs of partitions $v,w$ -- biaxial double $(\beta,q)$-Grothendieck
polynomials -- which specialize at $q=0$ and $v=1$ to double
$\beta$-Grothendieck polynomials from torus-equivariant connective K-theory.
Initially defined recursively via divided difference operators, our main result
is that these new polynomials arise as partition functions of solvable lattice
models. Moreover, the associated quantum group of the solvable model for
polynomials in $n$ pairs of variables is a Drinfeld twist of the
$U_q(\widehat{\mathfrak{sl}}_{n+1})$ $R$-matrix. By leveraging the resulting
Yang-Baxter equations of the lattice model, we show that these polynomials
simultaneously generalize double $\beta$-Grothendieck polynomials and dual
double $\beta$-Grothendieck polynomials for arbitrary permutations. We then use
properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's
Cauchy identity for $\beta$-Grothendieck polynomials, generalize it to a new
Cauchy identity for biaxial double $\beta$-Grothendieck polynomials, and prove
a new branching rule for double $\beta$-Grothendieck polynomials.
