This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer.
Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such that $$\sup_{x\in\mathcal{X}}\vert\vert\mathfrak{K}(x,\cdot)-\mathfrak{H}(x,\cdot)\vert\vert_\text{TV}<\epsilon$$ where $$\vert\vert\cdot\vert\vert_\text{TV}$$ denotes the total variation norm, is it possible that the Markov kernel $\mathfrak{K}$ is ergodic while the Markov kernel $\mathfrak{H}$ is not (i.e., is null recurrent or even transient)?
Consider the two Markov kernels on the finite state space $\{1,2,3,4,5\}$, expressed as stochastic matrices: $$ K_\epsilon=\left[\matrix{ 0&1-\epsilon&\epsilon&0&0\cr 1-\epsilon&\epsilon&0&0&0\cr \epsilon&0&1-2\epsilon&\epsilon&0\cr 0&0&\epsilon&0&1-\epsilon\cr 0&0&0&1-\epsilon&\epsilon\cr} \right] $$ and $$ H=\left[\matrix{ 0&1&0&0&0\cr 1&0&0&0&0\cr 0&0&1&0&0\cr 0&0&0&0&1\cr 0&0&0&1&0\cr} \right]. $$ The total variation distance between $K_\epsilon$ and $H$ is $2\epsilon$, and $K_\epsilon$ is ergodic (if $\epsilon>0$) while $H$ is not.