Let $E$ be a $\mathbb R$-Banach space, $d$ be a complete separable metric on $E$, $\delta$ denote the Dirac kernel on $(E,\mathcal B(E))$ and $\kappa$ be a Markov kernel on $(E,\mathcal B(E))$. Assume $$a:=\inf_{\substack{(x,\:y)\:\in\:E^2\\\left\|x\right\|_E,\:\left\|y\right\|_E\:\le\:c}}\sup_{\substack{\gamma\\\gamma\text{ is a coupling of }\delta_x\kappa\text{ and }\delta_y\kappa}}\gamma\left(\left\{\rho<\delta\right\}\right)>0\tag1$$ for all $c,\delta>0$.
Let $c,\delta>0$. How can we conclude that for all $(x,y)\in E^2$ with $\left\|x\right\|_E,\left\|y\right\|_E\le c$, there is a coupling $\gamma$ of $\delta_x\kappa$ and $\delta_y\kappa$ with $$\gamma\left(\left\{\rho\le\frac\delta2\right\}\right)>a?\tag2$$
Let $(x,y)\in E^2$ and $\varepsilon>0$. Then, clearly, there is a coupling $\gamma$ of $\delta_x\kappa$ and $\delta_y\kappa$ with $$\gamma\left(\left\{\rho<\delta\right\}\right)>a-\varepsilon.\tag3$$ Maybe $\delta$ should be replaced by $\delta/2$ in $(3)$ and clearly $$\gamma\left(\left\{\rho<\delta\right\}\right)=\gamma\left(\left\{\rho\le\frac\delta2\right\}\right)+\gamma\left(\left\{\rho\in\left(\frac\delta2,\delta\right)\right\}\right),\tag4$$ but I'm not able to obtain $(2)$ from that.
Assume that $E$ is Polish so that the set of couplings between any two probability measures on $E$ is compact w.r.t. the weak topology. Here weak topology refers to the topology of weak convergence for measures, that is $\gamma_n \to \gamma$ if $\int_E f d\gamma_n \to \int_E fd\gamma$ for every continuous bounded $f \colon E \to \mathbb{R}$.
Let $x,y \in E$ and set $F = \{\rho \leq \delta \}$ (the argument requires a closed set). Since $$\sup_{\gamma \text{ coupling of } \delta_x \kappa \text{ and } \delta_y \kappa} \gamma(F) \geq a,$$ for every $n$ there exists a coupling $\gamma_n$ such that $\gamma_n(F) \geq a-\frac{1}{n}$. By compactness we may assume that $\gamma_n$ converges weakly to some coupling $\gamma$. Then by the Portmanteau theorem we get $$\gamma(F) \geq \limsup_n \gamma_n(F) \geq a.$$
For the compactness of the set of couplings, check out lemma $0.13$ and Corollary $0.14$ here. The proof relies on the Prokhorov theorem.