Factorization of rings of integer-valued rational functions

$\text{DeclareMathOperator}\text{Int}Int\text{DeclareMathOperator}\text{IntR}Int{}^{\text{R}}$For a domain $D$, the ring $\text{Int}(D)$ of integer-valued polynomials over $D$ is atomic if $D$ satisfies the ascending chain condition on principal ideals. However, even for a discrete valuation domain $V$, the ring $\text{IntR}(V)$ of integer-valued rational functions over $V$ is antimatter. We introduce a family of atomic rings of integer-valued rational functions and study various factorization properties on these rings.