The notion of factorized $A_2$-Leonard pair is introduced. It is defined as a rank 2 Leonard pair, with actions in certain bases corresponding to the root system of the Weyl group $A_2$, and with some additional properties. The functions arising as entries of transition matrices are bivariate orthogonal polynomials (of Tratnik type) with bispectral properties. Examples of factorized $A_2$-Leonard pairs are constructed using classical Leonard pairs associated to families of orthogonal polynomials of the ($q$-)Askey scheme. The most general examples are associated to an intricate product of univariate ($q$-)Hahn and dual ($q$-)Hahn polynomials.