 Solution

For each auxiliary count $x$, the correctly normalized joint expression above is symmetric in $p_M,p_T$. Summing it over $x=0,\dots ,n$ preserves that symmetry, so these are exchangeable random variables. Their marginal distributions are therefore identical. Since $p_M$ was generated from a Beta distribution,The Beta-binomial exchangeable coupling changes the dependence, while leaving both prior distributions unchanged.