We obtain explicit double-contour representations for the correlation kernels of the discrete orthogonal ($β=1$) and symplectic ($β=4$) random matrix ensembles with Meixner, Charlier, and Krawtchouk weights. A single Cauchy--difference--quotient composition identity expresses all $β=1,4$ blocks in terms of the projection kernel and bounded rational multipliers. From these formulas we give short steepest-descent proofs of bulk and edge universality (sine/Airy/Bessel) with uniform error control, an explicit Meixner$\to$Laguerre hard-edge crossover, and a first $A^{-1}$ correction that follows directly from the integrable structure. Finally, we show that Archimedean Kuznetsov tests splice into the Pfaffian kernels by a bounded holomorphic symbol acting in the contour variable; the symbol enters only through the same Cauchy difference--quotient, so the leading sine/Airy/Bessel limits persist and the $A^{-1}$ term again comes from linearizing at the saddle(s).