Constant size commitment to a bivariate polynomial
I think the main issue for the bivariate polynomials is the coefficients of the polynomial $f(x,y)$. Since $\tau$ is a secret value then to calculate $g^{\alpha f(\tau)}$ we can use the public parameters. As you know, the users can obtain the coefficients based on a standard basis and then solve a multiplication.
$$g^{\alpha f(\tau)}=\prod_{i=0}^{n}{(g^{ \alpha \tau^i })}^{f_i}$$
For a univariate polynomial, the coefficients of $f_i$ are constant while if we consider bivariate polynomials then it is necessary to evaluate a second polynomial on the $\tau$. Precisely, the standard basis $(1,X,X^2,...,X^d)$ forms the univariate polynomials and the coefficients for a bivariate polynomial are changing by the second variable.