Largeness notions and polytime translation for $\forall \Sigma^0_3$-consequences of $\mathsf{RT}^2_2$

Le Hou\'erou, Patey and Yokoyama defined a parameterized version of $\alpha$-largeness to prove that $\mathsf{WKL}_0 + \mathsf{RT}^2_2$ is a $\forall \Sigma^0_3$-conservative extension of $\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2$, where $\forall \Sigma^0_3$ is the universal set-closure of the class of $\Sigma^0_3$-formulas. We introduce a variant of this notion of largeness and obtain polynomial bounds, using a tree partition theorem based on Milliken's tree theorem. Thanks to the framework of forcing interpretation, this yields that any proof of a $\forall \Sigma^0_3$-sentence in the theory $\mathsf{WKL}_0 + \mathsf{RT}^2_2$ can be translated into a proof in $\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2$ at the cost of a polynomial increase in size.