  We study the existence of $\text{SU}(2)^2$-invariant $G_2$-instantons on
$\mathbb{R}^4 \times S^3$ with the coclosed $G_2$-structures found on
[arXiv:2209.02761]. We find an explicit 1-parameter family of
$\text{SU}(2)^3$-invariant $G_2$-instantons on the trivial bundle on
$\mathbb{R}^4 \times S^3$ and study its ''bubbling'' behaviour. We prove the
existence a 1-parameter family on the identity bundle. We also provide
existence results for locally defined $\text{SU}(2)^2$-invariant
$G_2$-instantons.
